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URL: https://oeis.org/A391635

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A391635
Triangle read by rows: T(n,k) = Sum_{j=0..k} (-1)^(k-j) * binomial(k,j) * (5+j)^n.
2
1, 5, 1, 25, 11, 2, 125, 91, 36, 6, 625, 671, 434, 156, 24, 3125, 4651, 4380, 2550, 840, 120, 15625, 31031, 39962, 33540, 17760, 5400, 720, 78125, 201811, 341796, 388206, 294000, 142800, 40320, 5040, 390625, 1288991, 2796194, 4131036, 4198824, 2898000, 1300320, 342720, 40320
OFFSET
0,2
LINKS
José L. Cereceda, Sums of powers of integers and generalized Stirling numbers of the second kind, arXiv:2211.11648 [math.NT], 2022.
Donald E. Knuth, Johann Faulhaber and Sums of Powers, arXiv:math/9207222 [math.CA], 1992.
Petro Kolosov, Formulas for Sums of Powers, GitHub, 2025.
Petro Kolosov, Mathematica programs, GitHub, 2025.
FORMULA
Let F(t, n, k) = Sum_{j=0..k} (-1)^(k-j) * binomial(k,j) * (t+j)^n, then
F(0, n, k) = A131689(n,k),
F(1, n, k) = A028246(n-1,k-1),
F(2, n, k) = A038719(n,k),
F(3, n, k) = A391552(n,k),
F(4, n, k) = A391633(n,k),
F(5, n, k) = T(n,k) (this sequence).
T(n,k) = Sum_{j=0..n} binomial(5,j)*Stirling2(n,k+j)*(k+j)!.
Let S_m(n) be a sum of powers: S_m(n) = 1^m + 2^m + 3^m + ... + n^m, then
S_m(n) = Sum_{j=0..m} T(m,j) * ( C(n-4,j+1) + (-1)^j*C(j+4,j+1) ).
EXAMPLE
Triangle begins:
k= 0 1 2 3 4 5 6 7
------------------------------------------------------------------
n=0: 1;
n=1: 5, 1;
n=2: 25, 11, 2;
n=3: 125, 91, 36, 6;
n=4: 625, 671, 434, 156, 24;
n=5: 3125, 4651, 4380, 2550, 840, 120;
n=6: 15625, 31031, 39962, 33540, 17760, 5400, 720;
n=7: 78125, 201811, 341796, 388206, 294000, 142800, 40320, 5040;
...
MATHEMATICA
(* Prints the values of T(n, k) as triangle. *)
T[t_, n_, k_] := Sum[(-1)^(k - j)*Binomial[k, j]*(t + j)^n, {j, 0, n}];
Column[Table[T[5, n, k], {n, 0, 10}, {k, 0, n}]]
(* The formula in terms of Stirling2 numbers. *)
T2[t_, n_, k_] := Sum[Binomial[t, j]* StirlingS2[n, k + j]*(k + j)!, {j, 0, n}];
Column[Table[T2[5, n, k], {n, 0, 10}, {k, 0, n}]]
KEYWORD
nonn,tabl,easy,changed
AUTHOR
Petro Kolosov, Dec 14 2025
STATUS
approved