A396559 Triangle read by rows: T(n,k) = numerator(CF(n,k)), where CF(n,k) = (1/k!) * Sum_{j=0..k} (-1)^j * binomial(k,j) * (3+k/2-j)^n.

1, 3, 1, 9, 6, 1, 27, 109, 9, 1, 81, 111, 55, 12, 1, 243, 6841, 285, 185, 15, 1, 729, 12753, 1351, 585, 140, 18, 1, 2187, 372709, 6069, 53011, 1050, 791, 21, 1, 6561, 167943, 26335, 35049, 6951, 1722, 266, 24, 1, 19683, 19200241, 111645, 1417705, 42525, 104811, 2646, 345, 27, 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..54.

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. See p. 13.

Formula

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

F(0,2n,2k) = A387597(n,k)/k! = A269945(n,k);

F(1,2n,2k) = A392337(n,k)/k! = A394692(n,k);

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

F(3,2n,2k) = A395457(n,k);

numerators(F(0,n,k)) = A395862(n,k);

denominators(F(0,n,k)) = A370703(n,k);

numerators(F(1,n,k)) = A395860(n,k);

denominators(F(1,n,k)) = A395861(n,k);

numerators(F(2,n,k)) = A394466(n,k);

denominators(F(2,n,k)) = A395314(n,k);

numerators(F(3,n,k)) = T(n,k) (this sequence);

denominators(F(3,n,k)) = A395861(n,k).

G.f. for CF(n,k) is centered Newton's polynomial for f(s) = s^n: s^n = Sum_{k=0..n} CF(n,k) * (s-3)^[k], where (s-3)^[k] are central factorials.

Example

Triangle begins:
  [0]      1;
  [1]      3,        1;
  [2]      9,        6,      1;
  [3]     27,      109,      9,       1;
  [4]     81,      111,     55,      12,      1;
  [5]    243,     6841,    285,     185,     15,      1;
  [6]    729,    12753,   1351,     585,    140,     18,    1;
  [7]   2187,   372709,   6069,   53011,   1050,    791,   21,    1;
  [8]   6561,   167943,  26335,   35049,   6951,   1722,   266,   24,   1;
  [9]  19683, 19200241, 111645, 1417705,  42525, 104811,  2646,  345,  27,  1;
  [10] 59049, 34088703, 465751, 3474735, 246730, 365589, 22827, 3870, 435, 30, 1;

Mathematica

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

Crossrefs

Cf. A036969, A269945, A008957, A390029, A392337, A387597, A395456, A395457, A370703, A370705, A395860, A395861, A395314, A394466.

Sequence in context: A027465 A164942 A236420 * A187537 A246256 A157393

Adjacent sequences: A396556 A396557 A396558 * A396560 A396561 A396562

Keyword

nonn,easy,tabl,frac,changed

Author

Petro Kolosov, May 29 2026

Status

approved