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 A316349 #113 Aug 28 2025 10:29:33
%S A316349 31,60,30,512,540,150,2943,2160,420,10624,6000,900,29375,13500,1650,
%T A316349 68256,26460,2730,140287,47040,4200,263168,77760,6120,459999,121500,
%U A316349 8550,760000,181500,11550,1199231,261360,15180,1821312,365040,19500,2678143,496860,24570,3830624,661500,30450
%N A316349 Consider coefficients U(m,L,k) defined by the identity Sum_{k=1..L} Sum_{j=0..m} A302971(m,j)/A304042(m,j) * k^j * (T-k)^j = Sum_{k=0..m} (-1)^(m-k) * U(m,L,k) * T^k that holds for all positive integers L,m,T. This sequence gives 3-column table read by rows, where the n-th row lists coefficients U(2,n,k) for k = 0, 1, 2; n >= 1.
%C A316349 For L=T, the identity takes form T^(2m+1) = Sum_{k=0..m} (-1)^(m-k)*U(m,T,k)*T^k, which holds for all positive integers T and m.
%H A316349 Max Alekseyev, <a href="https://mathoverflow.net/q/309470">Derivation of the general formula for U(m,n,k)</a>, MathOverflow, 2018.
%H A316349 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 A316349 Petro Kolosov, <a href="https://kolosovpetro.github.io/pdf/OEIS_Um(n,k)_coefficients.pdf">More details on derivation of present sequence</a>.
%H A316349 Petro Kolosov, <a href="https://kolosovpetro.github.io/arxiv_1603_02468/identity_1_1_r_h_s.txt">Mathematica program</a>, verifies the identity T^(2m+1) = Sum_{k=0..m} (-1)^(m-k)*U(m,T,k)*T^k for m=0,1,...,12.
%H A316349 Petro Kolosov, <a href="https://kolosovpetro.github.io/pdf/HistoryAndOverviewOfPolynomialP.pdf">History and overview of the polynomial P_b^m(x)</a>, 2024.
%H A316349 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.
%F A316349 U(2,n,0) = 6*n^5 + 15*n^4 + 10*n^3; U(2,n,1) = 15*n^4 + 30*n^3 + 15*n^2; U(2,n,2) = 10*n^3 + 15*n^2 + 5*n. - _Max Alekseyev_, Sep 06 2018
%F A316349 From _Colin Barker_, Jul 06 2018: (Start)
%F A316349 G.f.: x*(31 + 60*x + 30*x^2 + 326*x^3 + 180*x^4 - 30*x^5 + 336*x^6 - 180*x^7 - 30*x^8 + 26*x^9 - 60*x^10 + 30*x^11 + x^12) / ((1 - x)^6*(1 + x + x^2)^6).
%F A316349 a(n) = 6*a(n-3) - 15*a(n-6) + 20*a(n-9) - 15*a(n-12) + 6*a(n-15) - a(n-18) for n>18. (End)
%F A316349 U(m,L,t) = (-1)^m * Sum_{k=1..L} Sum_{j=t..m} binomial(j,t) * R(m,j) * k^{2j-t} * (-1)^j, where m = 1, L >= 1 and R(m,j) = A302971(m,j)/A304042(m,j); after _Max Alekseyev_, see links.
%e A316349 column column column
%e A316349 L k=0 k=1 k=2
%e A316349 -- ------- ------- ------
%e A316349 1 31 60 30
%e A316349 2 512 540 150
%e A316349 3 2943 2160 420
%e A316349 4 10624 6000 900
%e A316349 5 29375 13500 1650
%e A316349 6 68256 26460 2730
%e A316349 7 140287 47040 4200
%e A316349 8 263168 77760 6120
%e A316349 9 459999 121500 8550
%e A316349 10 760000 181500 11550
%e A316349 11 1199231 261360 15180
%e A316349 12 1821312 365040 19500
%e A316349 ...
%t A316349 (* Define the R[n,k] := A302971(m,j)/A304042(m,j) *)
%t A316349 R[n_, k_] := 0
%t A316349 R[n_, k_] := (2 k + 1)*Binomial[2 k, k]*
%t A316349 Sum[R[n, j]*Binomial[j, 2 k + 1]*(-1)^(j - 1)/(j - k)*
%t A316349 BernoulliB[2 j - 2 k], {j, 2 k + 1, n}] /; 2 k + 1 <= n
%t A316349 R[n_, k_] := (2 n + 1)*Binomial[2 n, n] /; k == n;
%t A316349 (* Define the U(m,l,t) coefficients *)
%t A316349 U[m_, l_, t_] := (-1)^m Sum[Sum[Binomial[j, t] R[m,j] k^(2 j - t) (-1)^j, {j, t, m}], {k, 1, l}];
%t A316349 (* Define the value of the variable 'm', should be m = 2 for A316349 *)
%t A316349 m = 2;
%t A316349 (* Print first 10 rows of U(m,l,t) coefficients over l: 1 <= l <= 10 *)
%t A316349 Column[Table[U[m, l, t], {l, 1, 10}, {t, 0, m}]]
%Y A316349 The case m=1 is A320047.
%Y A316349 The case m=3 is A316387.
%Y A316349 Column k=0 is A316457.
%Y A316349 Column k=1 is A316458.
%Y A316349 Column k=2 is A316459.
%Y A316349 Cf. A302971, A304042, A287326, A300656, A300785.
%K A316349 nonn,tabf,changed
%O A316349 1,1
%A A316349 _Kolosov Petro_, Jun 29 2018
%E A316349 Edited by _Max Alekseyev_, Sep 06 2018