OFFSET
0,3
LINKS
P.-Y. Huang, S.-C. Liu, and Y.-N. Yeh, Congruences of Finite Summations of the Coefficients in certain Generating Functions, The Electronic Journal of Combinatorics, 21 (2014), #P2.45.
C. Jordan, Calculus of Finite Differences, Röttig and Romwalter, Budapest, 1939. [Annotated scans of pages 448-450 only]
Petro Kolosov, On the link between Binomial Theorem and Discrete Convolution of Power Function, arXiv:1603.02468 [math.NT], 2016-2020.
Petro Kolosov, Definition and table of values.
Petro Kolosov, An unusual identity for odd-powers, arXiv:2101.00227 [math.GM], 2021.
Petro Kolosov, Polynomial identity involving binomial theorem and Faulhaber's formula, 2023.
Petro Kolosov, History and overview of the polynomial P_b^m(x), 2024.
MathOverflow, Discussion of these coefficients, 2018.
FORMULA
Recurrence given by Max Alekseyev (see the MathOverflow link):
R(n, k) = 0 if k < 0 or k > n.
R(n, k) = (2k+1)*binomial(2k, k) if k = n.
R(n, k) = (2k+1)*binomial(2k, k)*Sum_{j=2k+1..n} R(n, j)*binomial(j, 2k+1)*(-1)^(j-1)/(j-k)*Bernoulli(2j-2k), otherwise.
T(n, k) = numerator(R(n, k)).
EXAMPLE
Triangle begins:
------------------------------------------------------------------------
k= 0 1 2 3 4 5 6 7 8
------------------------------------------------------------------------
n=0: 1;
n=1: 1, 6;
n=2: 1, 0, 30;
n=3: 1, -14, 0, 140;
n=4: 1, -120, 0, 0, 630;
n=5: 1, -1386, 660, 0, 0, 2772;
n=6: 1, -21840, 18018, 0, 0, 0, 12012;
n=7: 1, -450054, 491400, -60060, 0, 0, 0, 51480;
n=8: 1, -11880960, 15506040, -3712800, 0, 0, 0, 0, 218790;
MAPLE
R := proc(n, k) if k < 0 or k > n then return 0 fi; (2*k+1)*binomial(2*k, k);
if n = k then % else -%*add((-1)^j*R(n, j)*binomial(j, 2*k+1)*
bernoulli(2*j-2*k)/(j-k), j=2*k+1..n) fi end: T := (n, k) -> numer(R(n, k)):
seq(print(seq(T(n, k), k=0..n)), n=0..12);
# Numerical check that S(m, n) = n^(2*m+1):
S := (m, n) -> add(add(R(m, j)*(n-k)^j*k^j, j=0..m), k=0..n-1):
seq(seq(S(m, n) - n^(2*m+1), n=0..12), m=0..12); # Peter Luschny, Apr 30 2018
MATHEMATICA
R[n_, k_] := 0
R[n_, k_] := (2 k + 1)*Binomial[2 k, k]*
Sum[R[n, j]*Binomial[j, 2 k + 1]*(-1)^(j - 1)/(j - k)*
BernoulliB[2 j - 2 k], {j, 2 k + 1, n}] /; 2 k + 1 <= n
R[n_, k_] := (2 n + 1)*Binomial[2 n, n] /; k == n;
T[n_, k_] := Numerator[R[n, k]];
(* Print Fifteen Initial rows of Triangle A302971 *)
Column[ Table[ T[n, k], {n, 0, 15}, {k, 0, n}], Center]
PROG
(PARI) T(n, k) = if ((n>k) || (n<0), 0, if (k==n, (2*n+1)*binomial(2*n, n), if (2*n+1>k, 0, if (n==0, 1, (2*n+1)*binomial(2*n, n)*sum(j=2*n+1, k+1, T(j, k)*binomial(j, 2*n+1)*(-1)^(j-1)/(j-n)*bernfrac(2*j-2*n))))));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(numerator(T(k, n)), ", ")); print); \\ Michel Marcus, Apr 27 2018
CROSSREFS
Items of second row are the coefficients in the definition of A287326.
Items of third row are the coefficients in the definition of A300656.
Items of fourth row are the coefficients in the definition of A300785.
T(n,n) gives A002457(n).
Denominators of R(n,k) are shown in A304042.
Row sums return A000079(2n+1) - 1.
KEYWORD
AUTHOR
Kolosov Petro, Apr 16 2018
STATUS
approved