A302971 Triangle read by rows: T(n,k) is the numerator of R(n,k) defined implicitly by the identity Sum_{i=0..l-1} Sum_{j=0..m} R(m,j)*(l-i)^j*i^j = l^(2*m+1) holding for all l,m >= 0.

1, 1, 6, 1, 0, 30, 1, -14, 0, 140, 1, -120, 0, 0, 630, 1, -1386, 660, 0, 0, 2772, 1, -21840, 18018, 0, 0, 0, 12012, 1, -450054, 491400, -60060, 0, 0, 0, 51480, 1, -11880960, 15506040, -3712800, 0, 0, 0, 0, 218790, 1, -394788954, 581981400, -196409840, 8817900, 0, 0, 0, 0, 923780, 1, -16172552880, 26003271294, -10863652800, 1031151660, 0, 0, 0, 0, 0, 3879876

Offset

0,3

Links

Table of n, a(n) for n=0..65.

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.

Cf. A007318, A027641, A027642, A055012, A077028, A000146, A002882, A003245, A127187, A127188, A074909, A164555.

Sequence in context: A127573 A351110 A137388 * A114153 A119832 A166141

Adjacent sequences: A302968 A302969 A302970 * A302972 A302973 A302974

Keyword

sign,tabl,easy,frac

Author

Kolosov Petro, Apr 16 2018

Status

approved