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
Examples
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;
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, arXiv:1603.02468 [math.NT], 2016-2025.
- Petro Kolosov, Definition and table of values.
- Petro Kolosov, An unusual identity for odd-powers, arXiv:2101.00227 [math.GM], 2021.
- Petro Kolosov, 106.37 An unusual identity for odd-powers, The Mathematical Gazette, 2022.
- 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.
- Petro Kolosov, A novel proof of power rule in calculus, GitHub, 2024. See p. 5.
- Petro Kolosov, Odd-power identity via multiplication of certain matrices, GitHub, 2024. See p. 4.
- Petro Kolosov, An efficient method of spline approximation for power function, arXiv:2503.07618 [math.GM], 2025.
- Petro Kolosov, Discussion on coefficients of odd polynomial identity, GitHub, 2025.
- MathOverflow, Discussion of these coefficients, 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.
Programs
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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
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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] -
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
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)).