A395457 Triangle read by rows: T(n,k) = (1/(2k)!) * Sum_{j=0..2k} (-1)^j * binomial(2k,j) * (3+k-j)^(2n).

1, 9, 1, 81, 55, 1, 729, 1351, 140, 1, 6561, 26335, 6951, 266, 1, 59049, 465751, 246730, 22827, 435, 1, 531441, 7859215, 7508501, 1323652, 58542, 649, 1, 4782969, 129442951, 210766920, 63436373, 5146570, 128128, 910, 1, 43046721, 2104469695, 5652751651, 2734926558, 364558623, 16169582, 250978, 1220, 1

Offset

0,2

References

P. L. Butzer, K. Schmidt, E. L. Stark, and L. Vogt, Central factorial numbers; their main properties and some applications. Numerical Functional Analysis and Optimization, 10(5-6), (1989), 419-488.

J. Riordan, Combinatorial identities (Vol. 217), Wiley, New York, 1968.

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.

M. W. Coffey and M. C. Lettington, On Fibonacci Polynomial Expressions for Sums of m-th Powers, their implications for Faulhaber's Formula and some Theorems of Fermat, arXiv:1510.05402 [math.NT], 2015.

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

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

Petro Kolosov, Sums of powers of integers: A complete framework for closed formulas, Zenodo, 2026.

Formula

Let F(t, n, k) be generalized central factorial numbers of the second kind: F(t,n,k) = (1/(2k)!) * Sum_{j=0..2k} (-1)^j * binomial(2k,j) * (t+k-j)^(2n), then:

F(0,n,k) = A387597(n,k)/k! = A269945(n,k);

F(1,n,k) = A392337(n,k)/k! = A394692(n,k);

F(2,n,k) = A390029(n,k)/k! = A395456(n,k);

F(3,n,k) = T(n,k) (this sequence).

G.f. for T(n,k) is centered Newton's polynomial for f(s) = s^(2n): s^(2n) = Sum_{k=0..2n} T(2n,k) (s-3)^[k], where (s-3)^[k] are central factorials.

Example

Triangle begins:
  k=    0        1         2         3         4      5      6
  ----------------------------------------------------------------
  n=0:  1;
  n=1:  9,       1;
  n=2:  81,      55,       1;
  n=3:  729,     1351,     140,      1;
  n=4:  6561,    26335,    6951,     266,     1;
  n=5:  59049,   465751,   246730,   22827,   435,   1;
  n=6:  531441,  7859215,  7508501,  1323652, 58542, 649,   1;
  ...

Mathematica

T[t_, n_, k_] := 1/ k!* Sum[(-1)^j * Binomial[k, j] * (t + k/2 - j)^n, {j, 0, k}]; Column[Table[T[3, 2 n, 2 k], {n, 0, 10}, {k, 0, n}]]

Crossrefs

Cf. A036969, A269945, A008957, A390029, A392337, A387597, A394692, A395456.

Sequence in context: A347490 A038291 A075504 * A373628 A138342 A101678

Adjacent sequences: A395454 A395455 A395456 * A395458 A395459 A395460

Keyword

nonn,easy,tabl,changed

Author

Petro Kolosov, Apr 23 2026

Status

approved