A390029 Triangle read by rows: T(n,k) = Sum_{j=0..2k} (-1)^j * binomial(2k,j) * (2+k-j)^(2n).

1, 4, 2, 16, 50, 24, 64, 602, 1560, 720, 256, 6050, 40824, 90720, 40320, 1024, 57002, 818520, 4339440, 8467200, 3628800, 4096, 523250, 14676024, 151367040, 663949440, 1157587200, 479001600, 16384, 4750202, 249401880, 4564319760, 36785548800, 138032294400, 217945728000, 87178291200

Offset

0,2

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(2, n, k).

Example

Triangle begins:
  k=      0       1         2           3         4           5          6
  ---------------------------------------------------------------------------
  n=0:    1;
  n=1:    4,      2;
  n=2:   16,     50,       24;
  n=3:   64,    602,     1560,       720;
  n=4:  256,   6050,    40824,     90720,     40320;
  n=5: 1024,  57002,   818520,   4339440,   8467200,    3628800;
  n=6: 4096, 523250, 14676024, 151367040, 663949440, 1157587200, 479001600;
  ...

Mathematica

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

Crossrefs

Cf. A008957, A269945.

Sequence in context: A395930 A356569 A122749 * A189741 A381730 A303142

Adjacent sequences: A390026 A390027 A390028 * A390030 A390031 A390032

Keyword

nonn,tabl,easy

Author

Petro Kolosov, Jan 07 2026

Status

approved