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In Table of Integrals, Series, and Products. Seventh Edition. I.S. Gradshteyn and I.M. Ryzhik, there is 0.154.3 $$ \sum_{k=0}^N (-1)^k {N \choose k} k^{n-1} =0, N \geq n \geq 1; 0^0 ≡ 1 $$ 0.154.4 $$ \sum_{k=0}^n (-1)^k {n \choose k} k^{n} =(-1)^n n!, n \geq 0; 0^0 ≡ 1 $$

I would like to know, there is any generalization for 0.154.4 for even higer power of $k$? $$ \sum_{k=0}^n (-1)^k {n \choose k} k^{n+m} =?, n \geq 0; m\geq0; 0^0 ≡ 1 $$

In Table of Integrals, Series, and Products. Seventh Edition. I.S. Gradshteyn and I.M. Ryzhik, there is 0.154.3 $$ \sum_{k=0}^N (-1)^k {N \choose k} k^{n-1} =0, N \geq n \geq 1; 0^0 ≡ 1 $$ 0.154.4 $$ \sum_{k=0}^n (-1)^k {n \choose k} k^{n} =(-1)^n n!, n \geq 0; 0^0 ≡ 1 $$

I would like to know, there is any generalization for 0.154.4 for a general, higher power of $k$? $$ \sum_{k=0}^n (-1)^k {n \choose k} k^{n+m} =?, n \geq 0; m\geq0; 0^0 ≡ 1 $$

In Table of Integrals, Series, and Products. Seventh Edition. I.S. Gradshteyn and I.M. Ryzhik, there is 0.154.3 $$ \sum_{k=0}^N (-1)^k {N \choose k} k^{n-1} =0, N \geq n \geq 1; 0^0 ≡ 1 $$ 0.154.4 $$ \sum_{k=0}^N (-1)^k {n \choose k} k^{n} =(-1)^n n!, n \geq 0; 0^0 ≡ 1 $$

I would like to know, there is any generalization for 0.154.4 for even higer-power of $k$? $$ \sum_{k=0}^N (-1)^k {n \choose k} k^{n+m} =?, n \geq 0; m\geq0; 0^0 ≡ 1 $$

In Table of Integrals, Series, and Products. Seventh Edition. I.S. Gradshteyn and I.M. Ryzhik, there is 0.154.3 $$ \sum_{k=0}^N (-1)^k {N \choose k} k^{n-1} =0, N \geq n \geq 1; 0^0 ≡ 1 $$ 0.154.4 $$ \sum_{k=0}^n (-1)^k {n \choose k} k^{n} =(-1)^n n!, n \geq 0; 0^0 ≡ 1 $$

I would like to know, there is any generalization for 0.154.4 for even higer power of $k$? $$ \sum_{k=0}^n (-1)^k {n \choose k} k^{n+m} =?, n \geq 0; m\geq0; 0^0 ≡ 1 $$

?In Table of Integrals, Series, and Products. Seventh Edition. I.S. Gradshteyn and I.M. Ryzhik, there is 0.154.3 $$ \sum_{k=0}^N (-1)^k {N \choose k} k^{n-1} =0, N \geq n \geq 1; 0^0 ≡ 1 $$ 0.154.4 $$ \sum_{k=0}^N (-1)^k {n \choose k} k^{n} =(-1)^n n!, n \geq 0; 0^0 ≡ 1 $$

I would like to know, there is any generalization for 0.154.4 for even higer-power of $k$? $$ \sum_{k=0}^N (-1)^k {n \choose k} k^{n+m} =?, n \geq 0; m\geq0; 0^0 ≡ 1 $$

In Table of Integrals, Series, and Products. Seventh Edition. I.S. Gradshteyn and I.M. Ryzhik, there is 0.154.3 $$ \sum_{k=0}^N (-1)^k {N \choose k} k^{n-1} =0, N \geq n \geq 1; 0^0 ≡ 1 $$ 0.154.4 $$ \sum_{k=0}^N (-1)^k {n \choose k} k^{n} =(-1)^n n!, n \geq 0; 0^0 ≡ 1 $$

I would like to know, there is any generalization for 0.154.4 for even higer-power of $k$? $$ \sum_{k=0}^N (-1)^k {n \choose k} k^{n+m} =?, n \geq 0; m\geq0; 0^0 ≡ 1 $$