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A304042 Triangle read by rows: T(n,k) is the denominator 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.

%I A304042 #60 Jul 21 2025 00:49:28
%S A304042 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%T A304042 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,5,
%U A304042 1,1,1,5,1,1,1,1,1,1,1,3,1,3,1,1,1,1,1,1,1,1,1
%N A304042 Triangle read by rows: T(n,k) is the denominator 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 A304042 Antti Karttunen, <a href="/A304042/b304042.txt">Table of n, a(n) for n = 0..10439 (the first 144 rows of triangle)</a>
%H A304042 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 A304042 C. Jordan, <a href="https://oeis.org/A002457/a002457_1.pdf">Calculus of Finite Differences</a>, Budapest, 1939. [Annotated scans of pages 448-450 only]
%H A304042 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 A304042 Petro Kolosov, <a href="https://kolosovpetro.github.io/pdf/coefficients_of_polynomial.pdf">Definition and table of values</a>.
%H A304042 Petro Kolosov, <a href="https://arxiv.org/abs/2101.00227">An unusual identity for odd-powers</a>, arXiv:2101.00227 [math.GM], 2021.
%H A304042 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 A304042 Petro Kolosov, <a href="https://kolosovpetro.github.io/pdf/PolynomialIdentityInvolvingBTandFaulhaber.pdf">Polynomial identity involving binomial theorem and Faulhaber's formula</a>, 2023.
%H A304042 Petro Kolosov, <a href="https://kolosovpetro.github.io/pdf/HistoryAndOverviewOfPolynomialP.pdf">History and overview of the polynomial P_b^m(x)</a>, 2024.
%H A304042 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 A304042 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 A304042 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 A304042 Petro Kolosov, <a href="https://kolosovpetro.github.io/pdf/DiscussionOnCoefficientsOfOddPolynomialIdentity.pdf">Discussion on coefficients of odd polynomial identity</a>, GitHub, 2025.
%H A304042 MathOverflow, <a href="https://mathoverflow.net/q/297900">Discussion of these coefficients</a>, 2018.
%F A304042 Recurrence given by _Max Alekseyev_ (see the MathOverflow link):
%F A304042 R(n, k) = 0 if k < 0 or k > n.
%F A304042 R(n, k) = (2k+1)*binomial(2k, k) if k = n.
%F A304042 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 A304042 T(n, k) = denominator(R(n, k)).
%e A304042 Triangle begins:
%e A304042 -----------------------------------------------------
%e A304042 k=    0  1  2  3  4  5  6  7  8  9  10 11 12 13 14 15
%e A304042 -----------------------------------------------------
%e A304042 n=0:  1;
%e A304042 n=1:  1, 1;
%e A304042 n=2:  1, 1, 1;
%e A304042 n=3:  1, 1, 1, 1;
%e A304042 n=4:  1, 1, 1, 1, 1;
%e A304042 n=5:  1, 1, 1, 1, 1, 1;
%e A304042 n=6:  1, 1, 1, 1, 1, 1, 1;
%e A304042 n=7:  1, 1, 1, 1, 1, 1, 1, 1;
%e A304042 n=8:  1, 1, 1, 1, 1, 1, 1, 1, 1;
%e A304042 n=9:  1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%e A304042 n=10: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%e A304042 n=11: 1, 5, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1;
%e A304042 n=12: 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%e A304042 n=13: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%e A304042 n=14: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%e A304042 n=15: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%t A304042 R[n_, k_] := 0
%t A304042 R[n_, k_] := (2 k + 1)*Binomial[2 k, k]*
%t A304042    Sum[R[n, j]*Binomial[j, 2 k + 1]*(-1)^(j - 1)/(j - k)*
%t A304042    BernoulliB[2 j - 2 k], {j, 2 k + 1, n}] /; 2 k + 1 <= n
%t A304042 R[n_, k_] := (2 n + 1)*Binomial[2 n, n] /; k == n;
%t A304042 T[n_, k_] := Denominator[R[n, k]];
%t A304042 (* Print Fifteen Initial rows of Triangle A304042 *)
%t A304042 Column[ Table[ T[n, k], {n, 0, 15}, {k, 0, n}], Center]
%o A304042 (PARI)
%o A304042 up_to = 1274; \\ = binomial(50+1,2)-1
%o A304042 A304042aux(n, k) = if((k<0)||(k>n),0,(k+k+1)*binomial(2*k, k)*if(k==n,1,sum(j=k+k+1,n, A304042aux(n, j)*binomial(j, k+k+1)*((-1)^(j-1))/(j-k)*bernfrac(2*(j-k)))));
%o A304042 A304042tr(n, k) = denominator(A304042aux(n, k));
%o A304042 A304042list(up_to) = { my(v = vector(up_to), i=0); for(n=0,oo, for(k=0,n, if(i++ > up_to, return(v)); v[i] = A304042tr(n,k))); (v); };
%o A304042 v304042 = A304042list(1+up_to);
%o A304042 A304042(n) = v304042[1+n]; \\ _Antti Karttunen_, Nov 07 2018
%Y A304042 Numerators are shown in A302971.
%Y A304042 Cf. A287326, A300656, A300785, A007318, A027641, A027642, A055012, A077028, A000146, A002882, A003245, A127187, A127188, A074909, A164555.
%K A304042 nonn,tabl,frac
%O A304042 0,68
%A A304042 _Kolosov Petro_, May 05 2018