This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A302971 #170 Jul 21 2025 00:49:23
%S A302971 1,1,6,1,0,30,1,-14,0,140,1,-120,0,0,630,1,-1386,660,0,0,2772,1,
%T A302971 -21840,18018,0,0,0,12012,1,-450054,491400,-60060,0,0,0,51480,1,
%U A302971 -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
%N 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.
%H A302971 P.-Y. Huang, S.-C. Liu, and Y.-N. Yeh, <a href="https://doi.org/10.37236/3693">Congruences of Finite Summations of the Coefficients in certain Generating Functions</a>, The Electronic Journal of Combinatorics, 21 (2014), #P2.45.
%H A302971 C. Jordan, <a href="https://oeis.org/A002457/a002457_1.pdf">Calculus of Finite Differences</a>, Röttig and Romwalter, Budapest, 1939. [Annotated scans of pages 448-450 only]
%H A302971 Petro Kolosov, <a href="https://arxiv.org/abs/1603.02468">On the link between binomial theorem and discrete convolution</a>, arXiv:1603.02468 [math.NT], 2016-2025.
%H A302971 Petro Kolosov, <a href="https://kolosovpetro.github.io/pdf/coefficients_of_polynomial.pdf">Definition and table of values</a>.
%H A302971 Petro Kolosov, <a href="https://arxiv.org/abs/2101.00227">An unusual identity for odd-powers</a>, arXiv:2101.00227 [math.GM], 2021.
%H A302971 Petro Kolosov, <a href="https://doi.org/10.1017/mag.2022.129">106.37 An unusual identity for odd-powers</a>, The Mathematical Gazette, 2022.
%H A302971 Petro Kolosov, <a href="https://kolosovpetro.github.io/pdf/PolynomialIdentityInvolvingBTandFaulhaber.pdf">Polynomial identity involving binomial theorem and Faulhaber's formula</a>, 2023.
%H A302971 Petro Kolosov, <a href="https://kolosovpetro.github.io/pdf/HistoryAndOverviewOfPolynomialP.pdf">History and overview of the polynomial P_b^m(x)</a>, 2024.
%H A302971 Petro Kolosov, <a href="https://kolosovpetro.github.io/pdf/ANovelProofOfPowerRuleInCalculus.pdf">A novel proof of power rule in calculus</a>, GitHub, 2024. See p. 5.
%H A302971 Petro Kolosov, <a href="https://kolosovpetro.github.io/pdf/OddPowerIdentityViaMultiplicationOfCertainMatrices.pdf">Odd-power identity via multiplication of certain matrices</a>, GitHub, 2024. See p. 4.
%H A302971 Petro Kolosov, <a href="https://arxiv.org/abs/2503.07618">An efficient method of spline approximation for power function</a>, arXiv:2503.07618 [math.GM], 2025.
%H A302971 Petro Kolosov, <a href="https://kolosovpetro.github.io/pdf/DiscussionOnCoefficientsOfOddPolynomialIdentity.pdf">Discussion on coefficients of odd polynomial identity</a>, GitHub, 2025.
%H A302971 MathOverflow, <a href="https://mathoverflow.net/q/297900">Discussion of these coefficients</a>, 2018.
%F A302971 Recurrence given by _Max Alekseyev_ (see the MathOverflow link):
%F A302971 R(n, k) = 0 if k < 0 or k > n.
%F A302971 R(n, k) = (2k+1)*binomial(2k, k) if k = n.
%F A302971 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.
%F A302971 T(n, k) = numerator(R(n, k)).
%e A302971 Triangle begins:
%e A302971 ------------------------------------------------------------------------
%e A302971 k= 0 1 2 3 4 5 6 7 8
%e A302971 ------------------------------------------------------------------------
%e A302971 n=0: 1;
%e A302971 n=1: 1, 6;
%e A302971 n=2: 1, 0, 30;
%e A302971 n=3: 1, -14, 0, 140;
%e A302971 n=4: 1, -120, 0, 0, 630;
%e A302971 n=5: 1, -1386, 660, 0, 0, 2772;
%e A302971 n=6: 1, -21840, 18018, 0, 0, 0, 12012;
%e A302971 n=7: 1, -450054, 491400, -60060, 0, 0, 0, 51480;
%e A302971 n=8: 1, -11880960, 15506040, -3712800, 0, 0, 0, 0, 218790;
%p A302971 R := proc(n, k) if k < 0 or k > n then return 0 fi; (2*k+1)*binomial(2*k, k);
%p A302971 if n = k then % else -%*add((-1)^j*R(n, j)*binomial(j, 2*k+1)*
%p A302971 bernoulli(2*j-2*k)/(j-k), j=2*k+1..n) fi end: T := (n, k) -> numer(R(n, k)):
%p A302971 seq(print(seq(T(n, k), k=0..n)), n=0..12);
%p A302971 # Numerical check that S(m, n) = n^(2*m+1):
%p A302971 S := (m, n) -> add(add(R(m, j)*(n-k)^j*k^j, j=0..m), k=0..n-1):
%p A302971 seq(seq(S(m, n) - n^(2*m+1), n=0..12), m=0..12); # _Peter Luschny_, Apr 30 2018
%t A302971 R[n_, k_] := 0
%t A302971 R[n_, k_] := (2 k + 1)*Binomial[2 k, k]*
%t A302971 Sum[R[n, j]*Binomial[j, 2 k + 1]*(-1)^(j - 1)/(j - k)*
%t A302971 BernoulliB[2 j - 2 k], {j, 2 k + 1, n}] /; 2 k + 1 <= n
%t A302971 R[n_, k_] := (2 n + 1)*Binomial[2 n, n] /; k == n;
%t A302971 T[n_, k_] := Numerator[R[n, k]];
%t A302971 (* Print Fifteen Initial rows of Triangle A302971 *)
%t A302971 Column[ Table[ T[n, k], {n, 0, 15}, {k, 0, n}], Center]
%o A302971 (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))))));
%o A302971 tabl(nn) = for (n=0, nn, for (k=0, n, print1(numerator(T(k,n)), ", ")); print); \\ _Michel Marcus_, Apr 27 2018
%Y A302971 Items of second row are the coefficients in the definition of A287326.
%Y A302971 Items of third row are the coefficients in the definition of A300656.
%Y A302971 Items of fourth row are the coefficients in the definition of A300785.
%Y A302971 T(n,n) gives A002457(n).
%Y A302971 Denominators of R(n,k) are shown in A304042.
%Y A302971 Row sums return A000079(2n+1) - 1.
%Y A302971 Cf. A007318, A027641, A027642, A055012, A077028, A000146, A002882, A003245, A127187, A127188, A074909, A164555.
%K A302971 sign,tabl,easy,frac
%O A302971 0,3
%A A302971 _Kolosov Petro_, Apr 16 2018