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Let $a(n)$ be A110501 (i.e., unsigned Genocchi numbers (of first kind) of even index). Here

$$ a(n) = \sum\limits_{i=1}^{\left\lfloor\frac{n}{2}\right\rfloor}\binom{n}{2i}a(n-i)(-1)^{i-1}, \\ a(1) = 1 $$

Let $T(n, k)$ be A036969. Here

$$ T(n, k) = k^2T(n-1, k) + T(n-1, k-1), \\ T(n, 1) = 1 $$

Let $b(n)$ be an integer sequence such that

$$ b(n) = \sum\limits_{i=1}^{n-1} (i!)^2T(n-1, i)(-1)^{n-i-1}, \\ b(1) = 1 $$

I conjecture that $$b(n) = a(n).$$

Here is the PARI/GP program to check it numerically:

a(n) = if(n<1, 0, 2 * (-1)^n * (1 - 4^n) * bernfrac(2*n))
b_upto(n) = {my(x='x, v1, v2, A, B);
v1 = vector(n, i, 0); v1[1] = 1;
for(i=2, n+1,
if(i<(n+1), v2 = v1[i-1];
v1[i] = x*(v2 + v2' + x*v2''));
if(i>2, v2 = Vecrev(v1[i-1]/x);
A = 0; B = 1;
forstep(j=i-2, 1, -1,
A = A*(j+1)^2 + B*v2[j];
B *= -1);
v1[i-1] = A));
v1}
test(n) = b_upto(n) == vector(n, i, a(i))

Is there a way to prove it?

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1 Answer 1

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This is a known result. To quote from Richard Stanley's Enumerative Combinatorics, Volume 2, second edition, solution to problem 8(e) of Chapter 5, page 115: This is equivalent to a conjecture of J. M. Gandhi, Amer. Math. Monthly 77 (1970), 505–506. This conjecture was proved by L. Carlitz, K. Norske Vidensk. Selsk. Sk. 9 (1972), 1–4, and by J. Riordan and P. R. Stein, Discrete Math. 5 (1973), 381–388. A combinatorial proof of Gandhi's conjecture was given by J. Françon and G. Viennot, Discrete Math. 28 (1979), 21–35.

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  • $\begingroup$ Thank you for answer! What do you think about $$a(n) = \sum\limits_{i=1}^{n}i((i-1)!)^2T(n, i)(-1)^{n-i}.$$? $\endgroup$ Commented May 24 at 18:03
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    $\begingroup$ I don't know anything about this sum. $\endgroup$
    – Ira Gessel
    Commented May 25 at 3:16
  • $\begingroup$ I think your second sum may also be the unsigned Genocchi numbers (see Barsky, Dumont, Congruences Pour les Nombres de Genocchi de la 2e espèce, Study Group on Ultrametric Analysis (1981)). To shamelessly advertise a little, Michelle Wachs and I gave a unifying generalization of some of these identities via a generating function formula for the characteristic polynomials of some hyperplane arrangements. $\endgroup$
    – Alex Lazar
    Commented May 26 at 10:42

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