A391633 Triangle read by rows: T(n,k) = Sum_{j=0..k} (-1)^(k-j) * binomial(k,j) * (4+j)^n.

1, 4, 1, 16, 9, 2, 64, 61, 30, 6, 256, 369, 302, 132, 24, 1024, 2101, 2550, 1830, 720, 120, 4096, 11529, 19502, 20460, 13080, 4680, 720, 16384, 61741, 140070, 201726, 186480, 107520, 35280, 5040, 65536, 325089, 963902, 1832292, 2298744, 1900080, 997920, 302400, 40320

Offset

0,2

Links

Table of n, a(n) for n=0..44.

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.

Petro Kolosov, Newton's interpolation formula and sums of powers, Zenodo, 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) = T(n,k) (this sequence),

F(5, n, k) = A391635(n,k).

T(n,k) = Sum_{j=0..n} binomial(4,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-3,j+1) + (-1)^j*C(j+3,j+1) ).

Example

Triangle begins:
  k=    0      1      2       3       4       5       6      7
  ---------------------------------------------------------------
  n=0:  1;
  n=1:  4,     1;
  n=2:  16,    9,     2;
  n=3:  64,    61,    30,     6;
  n=4:  256,   369,   302,    132,    24;
  n=5:  1024,  2101,  2550,   1830,   720,    120;
  n=6:  4096,  11529, 19502,  20460,  13080,  4680,   720;
  n=7:  16384, 61741, 140070, 201726, 186480, 107520, 35280, 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[4, 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[4, n, k], {n, 0, 10}, {k, 0, n}]]

Crossrefs

Cf. A048993, A007318, A131689, A028246, A038719, A391552, A391635.

Sequence in context: A351419 A303054 A143496 * A143697 A374098 A272088

Adjacent sequences: A391630 A391631 A391632 * A391634 A391635 A391636

Keyword

nonn,tabl,easy

Author

Petro Kolosov, Dec 14 2025

Status

approved