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 A316387 #79 Aug 28 2025 10:29:04
%S A316387 125,406,420,140,9028,13818,7140,1260,110961,115836,41160,5040,684176,
%T A316387 545860,148680,14000,2871325,1858290,411180,31500,9402660,5124126,
%U A316387 955500,61740,25872833,12182968,1963920,109760,62572096,25945416,3684240,181440,136972701,50745870,6439860,283500,276971300,92745730,10639860,423500
%N A316387 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 4-column table read by rows, where the n-th row lists coefficients U(3,n,k) for k = 0, 1, 2, 3; n >= 1.
%C A316387 For T=L, 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 A316387 Max Alekseyev, <a href="https://mathoverflow.net/q/309470">Derivation of the general formula for U(m,n,k)</a>, MathOverflow, 2018.
%H A316387 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 A316387 Petro Kolosov, <a href="https://kolosovpetro.github.io/pdf/OEIS_Um(n,k)_coefficients.pdf">More details on derivation of present sequence</a>.
%H A316387 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 A316387 Petro Kolosov, <a href="https://kolosovpetro.github.io/pdf/HistoryAndOverviewOfPolynomialP.pdf">History and overview of the polynomial P_b^m(x)</a>, 2024.
%H A316387 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 A316387 U(3,n,0) = 20*n^7 + 70*n^6 + 70*n^5 - 28*n^3 - 7*n^2; U(3,n,1) = 70*n^6 + 210*n^5 + 175*n^4 - 42*n^2 - 7*n; U(3,n,2) = 84*n^5 + 210*n^4 + 140*n^3 - 14*n; U(3,n,3) = 35*n^4 + 70*n^3 + 35*n^2. - _Max Alekseyev_, Sep 06 2018
%F A316387 From _Colin Barker_, Jul 09 2018; corrected by _Max Alekseyev_, Sep 06 2018: (Start)
%F A316387 G.f.: x*(125 + 406*x + 420*x^2 + 140*x^3 + 8028*x^4 + 10570*x^5 + 3780*x^6 + 140*x^7 + 42237*x^8 + 16660*x^9 - 4200*x^10 - 1120*x^11 + 42272*x^12 - 16660*x^13 - 4200*x^14 + 1120*x^15 + 8007*x^16 - 10570*x^17 + 3780*x^18 - 140*x^19 + 132*x^20 - 406*x^21 + 420*x^22 - 140*x^23 - x^24) / ((1 - x)^8*(1 + x)^8*(1 + x^2)^8).
%F A316387 a(n) = 8*a(n-4) - 28*a(n-8) + 56*a(n-12) - 70*a(n-16) + 56*a(n-20) - 28*a(n-24) + 8*a(n-28) - a(n-32) for n>32. (End)
%F A316387 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. - _Kolosov Petro_, Oct 04 2018
%e A316387 column column column column
%e A316387 L k=0 k=1 k=2 k=3
%e A316387 -- ------------ ---------- ---------- -------
%e A316387 1 125 406 420 140
%e A316387 2 9028 13818 7140 1260
%e A316387 3 110961 115836 41160 5040
%e A316387 4 684176 545860 148680 14000
%e A316387 5 2871325 1858290 411180 31500
%e A316387 6 9402660 5124126 955500 61740
%e A316387 7 25872833 12182968 1963920 109760
%e A316387 8 62572096 25945416 3684240 181440
%e A316387 9 136972701 50745870 6439860 283500
%e A316387 10 276971300 92745730 10639860 423500
%e A316387 11 524988145 160386996 16789080 609840
%e A316387 12 943023888 264896268 25498200 851760
%e A316387 13 1618774781 420839146 37493820 1159340
%e A316387 14 2672907076 646725030 53628540 1543500
%e A316387 15 4267591425 965662320 74891040 2016000
%e A316387 16 6616398080 1406064016 102416160 2589440
%e A316387 17 9995653693 2002403718 137494980 3277260
%e A316387 18 14757360516 2796022026 181584900 4093740
%e A316387 19 21343778801 3835983340 236319720 5054000
%e A316387 20 30303773200 5179983060 303519720 6174000
%e A316387 ...
%t A316387 (* Define the R[n,k] := A302971(m,j)/A304042(m,j) *)
%t A316387 R[n_, k_] := 0
%t A316387 R[n_, k_] := (2 k + 1)*Binomial[2 k, k]*
%t A316387 Sum[R[n, j]*Binomial[j, 2 k + 1]*(-1)^(j - 1)/(j - k)*
%t A316387 BernoulliB[2 j - 2 k], {j, 2 k + 1, n}] /; 2 k + 1 <= n
%t A316387 R[n_, k_] := (2 n + 1)*Binomial[2 n, n] /; k == n;
%t A316387 (* Define the U(m,l,t) coefficients *)
%t A316387 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 A316387 (* Define the value of the variable 'm' *)
%t A316387 m = 3;
%t A316387 (* Print first 10 rows of U(m,l,t) coefficients for 'm' defined above *)
%t A316387 Column[Table[U[m, l, t], {l, 1, 10}, {t, 0, m}]]
%Y A316387 The case m=1 is A320047.
%Y A316387 The case m=2 is A316349.
%Y A316387 Column k=0 is A317981.
%Y A316387 Column k=1 is A317982.
%Y A316387 Column k=2 is A317983.
%Y A316387 Column k=3 is A317984.
%Y A316387 Cf. A287326, A300656, A300785, A302971, A304042.
%K A316387 nonn,tabf,changed
%O A316387 1,1
%A A316387 _Kolosov Petro_, Jul 01 2018
%E A316387 Edited by _Max Alekseyev_, Sep 06 2018