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Carlo Beenakker
Carlo Beenakker
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Inspection of the Mathematica output

In: Table[Sum[Binomial[s2 + s1 - 1, s2]* Binomial[n - s1 - 2, n - 1 - s1 - s2], {s2, 1, n - 3}, {s1, 1,  n - 2 - s2}], {n, 4, 20}]
Out: {1, 5, 18, 57, 169, 482, 1341, 3669, 9922, 26609, 70929, 188226, 497845, 1313501, 3459042, 9096393, 23895673}

indicates it's the series OEIS A258109 $$a_n=\frac{1}{5} 2^{-n-1} \left[\left(\sqrt{5}+5\right) \left(3-\sqrt{5}\right)^n-5\ 4^n-\left(\sqrt{5}-5\right) \left(\sqrt{5}+3\right)^n\right].$$

Here is a derivation:

$$\sum\limits_{(s_1,s_2,s_3)}\prod_{i=1}^3\binom{s_i+s_{i-1}-1}{s_i}=$$ $$=\sum_{s_1=1}^{n-2-s_2}\binom{s_1+s_2-1}{s_2} \binom{n-s_1-2}{n-s_1-s_2-1}$$ $$=2-2^{n-2}+\sum_{s_2=1}^{n-3}\binom{n-3}{n-s_2-2}\frac{\Gamma (3-n) \Gamma (-2 s_2) }{\Gamma (1-s_2) \Gamma (-n-s_2+2)}$$ $$=2-2^{n-2}+\frac{1}{2}\sum_{s_2=1}^{n-3}\binom{n-3}{n-s_2-2}\frac{\Gamma (n+s_2-1) \Gamma (s_2) }{\Gamma (1+2s_2) \Gamma (n-2)}$$ $$=1-2^{n-2}+\frac{\cosh \left((2 n-3) \text{csch}^{-1}(2)\right)}{\sqrt{5}}$$ I must have missed some factors, the final result does not match, will try to fix that later this evening.

Inspection of the Mathematica output

In: Table[Sum[Binomial[s2 + s1 - 1, s2]* Binomial[n - s1 - 2, n - 1 - s1 - s2], {s2, 1, n - 3}, {s1, 1,  n - 2 - s2}], {n, 4, 20}]
Out: {1, 5, 18, 57, 169, 482, 1341, 3669, 9922, 26609, 70929, 188226, 497845, 1313501, 3459042, 9096393, 23895673}

indicates it's the series OEIS A258109 $$a_n=\frac{1}{5} 2^{-n-1} \left[\left(\sqrt{5}+5\right) \left(3-\sqrt{5}\right)^n-5\ 4^n-\left(\sqrt{5}-5\right) \left(\sqrt{5}+3\right)^n\right].$$

Here is a derivation:

$$\sum\limits_{(s_1,s_2,s_3)}\prod_{i=1}^3\binom{s_i+s_{i-1}-1}{s_i}=$$ $$=\sum_{s_1=1}^{n-2-s_2}\binom{s_1+s_2-1}{s_2} \binom{n-s_1-2}{n-s_1-s_2-1}$$ $$=2-2^{n-2}+\sum_{s_2=1}^{n-3}\binom{n-3}{n-s_2-2}\frac{\Gamma (3-n) \Gamma (-2 s_2) }{\Gamma (1-s_2) \Gamma (-n-s_2+2)}$$ $$=2-2^{n-2}+\frac{1}{2}\sum_{s_2=1}^{n-3}\binom{n-3}{n-s_2-2}\frac{\Gamma (n+s_2-1) \Gamma (s_2) }{\Gamma (1+2s_2) \Gamma (n-2)}$$ $$=1-2^{n-2}+\frac{\cosh \left((2 n-3) \text{csch}^{-1}(2)\right)}{\sqrt{5}}$$ I must have missed some factors, the final result does not match, will try to fix that later this evening.

Inspection of the Mathematica output

In: Table[Sum[Binomial[s2 + s1 - 1, s2]* Binomial[n - s1 - 2, n - 1 - s1 - s2], {s2, 1, n - 3}, {s1, 1,  n - 2 - s2}], {n, 4, 20}]
Out: {1, 5, 18, 57, 169, 482, 1341, 3669, 9922, 26609, 70929, 188226, 497845, 1313501, 3459042, 9096393, 23895673}

indicates it's the series OEIS A258109 $$a_n=\frac{1}{5} 2^{-n-1} \left[\left(\sqrt{5}+5\right) \left(3-\sqrt{5}\right)^n-5\ 4^n-\left(\sqrt{5}-5\right) \left(\sqrt{5}+3\right)^n\right].$$

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Carlo Beenakker
Carlo Beenakker
  • 188.2k
  • 18
  • 448
  • 651

Inspection of the Mathematica output

In: Table[Sum[Binomial[s2 + s1 - 1, s2]* Binomial[n - s1 - 2, n - 1 - s1 - s2], {s2, 1, n - 3}, {s1, 1,  n - 2 - s2}], {n, 4, 20}]
Out: {1, 5, 18, 57, 169, 482, 1341, 3669, 9922, 26609, 70929, 188226, 497845, 1313501, 3459042, 9096393, 23895673}

indicates it's the series OEIS A258109 $$a_n=\frac{1}{5} 2^{-n-1} \left[\left(\sqrt{5}+5\right) \left(3-\sqrt{5}\right)^n-5\ 4^n-\left(\sqrt{5}-5\right) \left(\sqrt{5}+3\right)^n\right].$$

Here is a derivation:

$$\sum\limits_{(s_1,s_2,s_3)}\prod_{i=1}^3\binom{s_i+s_{i-1}-1}{s_i}=$$ $$=\sum_{s_1=1}^{n-2-s_2}\binom{s_1+s_2-1}{s_2} \binom{n-s_1-2}{n-s_1-s_2-1}$$ $$=2-2^{n-2}+\sum_{s_2=1}^{n-3}\binom{n-3}{n-s_2-2}\frac{\Gamma (3-n) \Gamma (-2 s_2) }{\Gamma (1-s_2) \Gamma (-n-s_2+2)}$$ $$=2-2^{n-2}+\frac{1}{2}\sum_{s_2=1}^{n-3}\binom{n-3}{n-s_2-2}\frac{\Gamma (n+s_2-1) \Gamma (s_2) }{\Gamma (1+2s_2) \Gamma (n-2)}$$ $$=1-2^{n-2}+\frac{\cosh \left((2 n-3) \text{csch}^{-1}(2)\right)}{\sqrt{5}}$$ I must have missed some factors, the final result does not match, will try to fix that later. I must have missed some factors, the final result does not match, will try to fix that later this evening.

Inspection of the Mathematica output

In: Table[Sum[Binomial[s2 + s1 - 1, s2]* Binomial[n - s1 - 2, n - 1 - s1 - s2], {s2, 1, n - 3}, {s1, 1,  n - 2 - s2}], {n, 4, 20}]
Out: {1, 5, 18, 57, 169, 482, 1341, 3669, 9922, 26609, 70929, 188226, 497845, 1313501, 3459042, 9096393, 23895673}

indicates it's the series OEIS A258109 $$a_n=\frac{1}{5} 2^{-n-1} \left[\left(\sqrt{5}+5\right) \left(3-\sqrt{5}\right)^n-5\ 4^n-\left(\sqrt{5}-5\right) \left(\sqrt{5}+3\right)^n\right].$$

Here is a derivation:

$$\sum\limits_{(s_1,s_2,s_3)}\prod_{i=1}^3\binom{s_i+s_{i-1}-1}{s_i}=$$ $$=\sum_{s_1=1}^{n-2-s_2}\binom{s_1+s_2-1}{s_2} \binom{n-s_1-2}{n-s_1-s_2-1}$$ $$=2-2^{n-2}+\sum_{s_2=1}^{n-3}\binom{n-3}{n-s_2-2}\frac{\Gamma (3-n) \Gamma (-2 s_2) }{\Gamma (1-s_2) \Gamma (-n-s_2+2)}$$ $$=2-2^{n-2}+\frac{1}{2}\sum_{s_2=1}^{n-3}\binom{n-3}{n-s_2-2}\frac{\Gamma (n+s_2-1) \Gamma (s_2) }{\Gamma (1+2s_2) \Gamma (n-2)}$$ $$=1-2^{n-2}+\frac{\cosh \left((2 n-3) \text{csch}^{-1}(2)\right)}{\sqrt{5}}$$ I must have missed some factors, the final result does not match, will try to fix that later.

Inspection of the Mathematica output

In: Table[Sum[Binomial[s2 + s1 - 1, s2]* Binomial[n - s1 - 2, n - 1 - s1 - s2], {s2, 1, n - 3}, {s1, 1,  n - 2 - s2}], {n, 4, 20}]
Out: {1, 5, 18, 57, 169, 482, 1341, 3669, 9922, 26609, 70929, 188226, 497845, 1313501, 3459042, 9096393, 23895673}

indicates it's the series OEIS A258109 $$a_n=\frac{1}{5} 2^{-n-1} \left[\left(\sqrt{5}+5\right) \left(3-\sqrt{5}\right)^n-5\ 4^n-\left(\sqrt{5}-5\right) \left(\sqrt{5}+3\right)^n\right].$$

Here is a derivation:

$$\sum\limits_{(s_1,s_2,s_3)}\prod_{i=1}^3\binom{s_i+s_{i-1}-1}{s_i}=$$ $$=\sum_{s_1=1}^{n-2-s_2}\binom{s_1+s_2-1}{s_2} \binom{n-s_1-2}{n-s_1-s_2-1}$$ $$=2-2^{n-2}+\sum_{s_2=1}^{n-3}\binom{n-3}{n-s_2-2}\frac{\Gamma (3-n) \Gamma (-2 s_2) }{\Gamma (1-s_2) \Gamma (-n-s_2+2)}$$ $$=2-2^{n-2}+\frac{1}{2}\sum_{s_2=1}^{n-3}\binom{n-3}{n-s_2-2}\frac{\Gamma (n+s_2-1) \Gamma (s_2) }{\Gamma (1+2s_2) \Gamma (n-2)}$$ $$=1-2^{n-2}+\frac{\cosh \left((2 n-3) \text{csch}^{-1}(2)\right)}{\sqrt{5}}$$ I must have missed some factors, the final result does not match, will try to fix that later this evening.

added 132 characters in body
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Carlo Beenakker
Carlo Beenakker
  • 188.2k
  • 18
  • 448
  • 651

Inspection of the Mathematica output

In: Table[Sum[Binomial[s2 + s1 - 1, s2]* Binomial[n - s1 - 2, n - 1 - s1 - s2], {s2, 1, n - 3}, {s1, 1,  n - 2 - s2}], {n, 4, 20}]
Out: {1, 5, 18, 57, 169, 482, 1341, 3669, 9922, 26609, 70929, 188226, 497845, 1313501, 3459042, 9096393, 23895673}

indicates it's the series OEIS A258109 $$a_n=\frac{1}{5} 2^{-n-1} \left[\left(\sqrt{5}+5\right) \left(3-\sqrt{5}\right)^n-5\ 4^n-\left(\sqrt{5}-5\right) \left(\sqrt{5}+3\right)^n\right].$$

As a first step towardsHere is a proof, I note that given $ s_1+s_2+s_3=n-1$, $s_1,s_2,s_3\geq 1$,derivation:

$$\sum\limits_{(s_1,s_2,s_3)}\prod_{i=1}^3\binom{s_i+s_{i-1}-1}{s_i}=$$ $$=\sum_{s_1=1}^{n-2-s_2}\binom{s_1+s_2-1}{s_2} \binom{n-s_1-2}{n-s_1-s_2-1}$$ $$=2-2^{n-2}+\sum_{s_2=1}^{n-3}\binom{n-3}{n-s_2-2}\frac{\Gamma (3-n) \Gamma (-2 s_2) }{\Gamma (1-s_2) \Gamma (-n-s_2+2)}$$ $$=2-2^{n-2}+\frac{1}{2}\sum_{s_2=1}^{n-3}\binom{n-3}{n-s_2-2}\frac{\Gamma (n+s_2-1) \Gamma (s_2) }{\Gamma (1+2s_2) \Gamma (n-2)}$$ $$=1-2^{n-2}+\frac{\cosh \left((2 n-3) \text{csch}^{-1}(2)\right)}{\sqrt{5}}$$ I must have missed some factors, the final result does not match, will try to fix that later.

Inspection of the Mathematica output

In: Table[Sum[Binomial[s2 + s1 - 1, s2]* Binomial[n - s1 - 2, n - 1 - s1 - s2], {s2, 1, n - 3}, {s1, 1,  n - 2 - s2}], {n, 4, 20}]
Out: {1, 5, 18, 57, 169, 482, 1341, 3669, 9922, 26609, 70929, 188226, 497845, 1313501, 3459042, 9096393, 23895673}

indicates it's the series OEIS A258109 $$a_n=\frac{1}{5} 2^{-n-1} \left[\left(\sqrt{5}+5\right) \left(3-\sqrt{5}\right)^n-5\ 4^n-\left(\sqrt{5}-5\right) \left(\sqrt{5}+3\right)^n\right].$$

As a first step towards a proof, I note that given $ s_1+s_2+s_3=n-1$, $s_1,s_2,s_3\geq 1$,

$$\sum\limits_{(s_1,s_2,s_3)}\prod_{i=1}^3\binom{s_i+s_{i-1}-1}{s_i}=$$ $$=\sum_{s_1=1}^{n-2-s_2}\binom{s_1+s_2-1}{s_2} \binom{n-s_1-2}{n-s_1-s_2-1}$$ $$=2-2^{n-2}+\sum_{s_2=1}^{n-3}\binom{n-3}{n-s_2-2}\frac{\Gamma (3-n) \Gamma (-2 s_2) }{\Gamma (1-s_2) \Gamma (-n-s_2+2)}$$ $$=2-2^{n-2}+\frac{1}{2}\sum_{s_2=1}^{n-3}\binom{n-3}{n-s_2-2}\frac{\Gamma (n+s_2-1) \Gamma (s_2) }{\Gamma (1+2s_2) \Gamma (n-2)}$$

Inspection of the Mathematica output

In: Table[Sum[Binomial[s2 + s1 - 1, s2]* Binomial[n - s1 - 2, n - 1 - s1 - s2], {s2, 1, n - 3}, {s1, 1,  n - 2 - s2}], {n, 4, 20}]
Out: {1, 5, 18, 57, 169, 482, 1341, 3669, 9922, 26609, 70929, 188226, 497845, 1313501, 3459042, 9096393, 23895673}

indicates it's the series OEIS A258109 $$a_n=\frac{1}{5} 2^{-n-1} \left[\left(\sqrt{5}+5\right) \left(3-\sqrt{5}\right)^n-5\ 4^n-\left(\sqrt{5}-5\right) \left(\sqrt{5}+3\right)^n\right].$$

Here is a derivation:

$$\sum\limits_{(s_1,s_2,s_3)}\prod_{i=1}^3\binom{s_i+s_{i-1}-1}{s_i}=$$ $$=\sum_{s_1=1}^{n-2-s_2}\binom{s_1+s_2-1}{s_2} \binom{n-s_1-2}{n-s_1-s_2-1}$$ $$=2-2^{n-2}+\sum_{s_2=1}^{n-3}\binom{n-3}{n-s_2-2}\frac{\Gamma (3-n) \Gamma (-2 s_2) }{\Gamma (1-s_2) \Gamma (-n-s_2+2)}$$ $$=2-2^{n-2}+\frac{1}{2}\sum_{s_2=1}^{n-3}\binom{n-3}{n-s_2-2}\frac{\Gamma (n+s_2-1) \Gamma (s_2) }{\Gamma (1+2s_2) \Gamma (n-2)}$$ $$=1-2^{n-2}+\frac{\cosh \left((2 n-3) \text{csch}^{-1}(2)\right)}{\sqrt{5}}$$ I must have missed some factors, the final result does not match, will try to fix that later.

added 132 characters in body
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Carlo Beenakker
Carlo Beenakker
  • 188.2k
  • 18
  • 448
  • 651
added 269 characters in body
Source Link
Carlo Beenakker
Carlo Beenakker
  • 188.2k
  • 18
  • 448
  • 651
added 269 characters in body
Source Link
Carlo Beenakker
Carlo Beenakker
  • 188.2k
  • 18
  • 448
  • 651
added 307 characters in body
Source Link
Carlo Beenakker
Carlo Beenakker
  • 188.2k
  • 18
  • 448
  • 651
Source Link
Carlo Beenakker
Carlo Beenakker
  • 188.2k
  • 18
  • 448
  • 651