A396559 - OEIS

%I #8 Jun 23 2026 15:39:13

%S 1,3,1,9,6,1,27,109,9,1,81,111,55,12,1,243,6841,285,185,15,1,729,

%T 12753,1351,585,140,18,1,2187,372709,6069,53011,1050,791,21,1,6561,

%U 167943,26335,35049,6951,1722,266,24,1,19683,19200241,111645,1417705,42525,104811,2646,345,27,1

%N 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.

%D 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.

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

%H José L. Cereceda, <a href="https://arxiv.org/abs/2211.11648">Sums of powers of integers and generalized Stirling numbers of the second kind</a>, arXiv:2211.11648 [math.NT], 2022.

%H M. W. Coffey and M. C. Lettington, <a href="https://arxiv.org/abs/1510.05402">On Fibonacci Polynomial Expressions for Sums of m-th Powers, their implications for Faulhaber's Formula and some Theorems of Fermat</a>, arXiv:1510.05402 [math.NT], 2015.

%H Donald E. Knuth, <a href="https://arxiv.org/abs/math/9207222">Johann Faulhaber and Sums of Powers</a>, arXiv:math/9207222 [math.CA], 1992.

%H Petro Kolosov, <a href="https://doi.org/10.5281/zenodo.18096789">Sums of powers via central finite differences and Newton's formula</a>, Zenodo, 2026.

%H Petro Kolosov, <a href="https://doi.org/10.5281/zenodo.20802514">Sums of powers of integers: A complete framework for closed formulas</a>, Zenodo, 2026. See p. 13.

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

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

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

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

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

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

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

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

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

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

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

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

%F 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.

%e Triangle begins:

%e   [0]      1;

%e   [1]      3,        1;

%e   [2]      9,        6,      1;

%e   [3]     27,      109,      9,       1;

%e   [4]     81,      111,     55,      12,      1;

%e   [5]    243,     6841,    285,     185,     15,      1;

%e   [6]    729,    12753,   1351,     585,    140,     18,    1;

%e   [7]   2187,   372709,   6069,   53011,   1050,    791,   21,    1;

%e   [8]   6561,   167943,  26335,   35049,   6951,   1722,   266,   24,   1;

%e   [9]  19683, 19200241, 111645, 1417705,  42525, 104811,  2646,  345,  27,  1;

%e   [10] 59049, 34088703, 465751, 3474735, 246730, 365589, 22827, 3870, 435, 30, 1;

%t 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}]]

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

%K nonn,easy,tabl,frac,changed

%O 0,2

%A _Petro Kolosov_, May 29 2026