A387597 Triangle read by rows: T(n,k) = Sum_{j=0..2k} (-1)^j * binomial(2k,j) * (k-j)^(2n).

1, 0, 2, 0, 2, 24, 0, 2, 120, 720, 0, 2, 504, 10080, 40320, 0, 2, 2040, 105840, 1209600, 3628800, 0, 2, 8184, 1013760, 25280640, 199584000, 479001600, 0, 2, 32760, 9369360, 461260800, 7264857600, 43589145600, 87178291200

Offset

0,3

Links

Table of n, a(n) for n=0..35.

Donald E. Knuth, Johann Faulhaber and Sums of Powers, arXiv:math/9207222 [math.CA], 1992.

Petro Kolosov, Mathematica programs, GitHub, 2026.

Petro Kolosov, Sums of powers via central finite differences and Newton's formula, Zenodo, 2026.

Formula

Let F(t,n,k) be a 2k-order central finite difference of power t^(2n): F(t,n,k) = Sum_{j=0..2k} (-1)^j * binomial(2k,j) * (t+k-j)^(2n). Then:

T(n,k) = F(0, n, k).

T(n,k) = (2*k)! * A269945(n,k).

Example

Triangle begins:
  k=   0  1     2        3         4          5          6
  ----------------------------------------------------------
  n=0: 1;
  n=1: 0, 2;
  n=2: 0, 2,   24;
  n=3: 0, 2,  120,     720;
  n=4: 0, 2,  504,   10080,    40320;
  n=5: 0, 2, 2040,  105840,  1209600,   3628800;
  n=6: 0, 2, 8184, 1013760, 25280640, 199584000, 479001600;
  ...

Mathematica

T[t_, n_, k_] := Sum[(-1)^j * Binomial[2 k, j]* (t + k - j)^(2 n), {j, 0, 2 k}]; Table[T[0, n, k], {n, 0, 10}, {k, 0, n}] // Flatten

Crossrefs

Cf. A008957, A269945.

Sequence in context: A077183 A101030 A093857 * A056949 A346235 A281326

Adjacent sequences: A387594 A387595 A387596 * A387598 A387599 A387600

Keyword

nonn,tabl,easy

Author

Petro Kolosov, Jan 07 2026

Status

approved