A392337 Triangle read by rows: T(n,k) = Sum_{j=0..2k} (-1)^j * binomial(2k,j) * (1+k-j)^(2n).

1, 1, 2, 1, 14, 24, 1, 62, 480, 720, 1, 254, 5544, 30240, 40320, 1, 1022, 54960, 710640, 3024000, 3628800, 1, 4094, 515064, 13654080, 125072640, 439084800, 479001600, 1, 16382, 4717440, 239999760, 4093689600, 29059430400, 87178291200, 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(1, n, k).

Example

Triangle begins:
  k=   0     1       2         3          4          5          6
  -----------------------------------------------------------------
  n=0: 1;
  n=1: 1,    2;
  n=2: 1,   14,     24;
  n=3: 1,   62,    480,      720;
  n=4: 1,  254,   5544,    30240,     40320;
  n=5: 1, 1022,  54960,   710640,   3024000,   3628800;
  n=6: 1, 4094, 515064, 13654080, 125072640, 439084800, 479001600;
  ...

Mathematica

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

Crossrefs

Cf. A008957, A269945.

Sequence in context: A245733 A276851 A080346 * A216445 A124026 A106204

Adjacent sequences: A392334 A392335 A392336 * A392338 A392339 A392340

Keyword

nonn,tabl,easy

Author

Petro Kolosov, Jan 07 2026

Status

approved