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.
125, 406, 420, 140, 9028, 13818, 7140, 1260, 110961, 115836, 41160, 5040, 684176, 545860, 148680, 14000, 2871325, 1858290, 411180, 31500, 9402660, 5124126, 955500, 61740, 25872833, 12182968, 1963920, 109760, 62572096, 25945416, 3684240, 181440, 136972701, 50745870, 6439860, 283500, 276971300, 92745730, 10639860, 423500
Offset: 1
Examples
column column column column L k=0 k=1 k=2 k=3 -- ------------ ---------- ---------- ------- 1 125 406 420 140 2 9028 13818 7140 1260 3 110961 115836 41160 5040 4 684176 545860 148680 14000 5 2871325 1858290 411180 31500 6 9402660 5124126 955500 61740 7 25872833 12182968 1963920 109760 8 62572096 25945416 3684240 181440 9 136972701 50745870 6439860 283500 10 276971300 92745730 10639860 423500 11 524988145 160386996 16789080 609840 12 943023888 264896268 25498200 851760 13 1618774781 420839146 37493820 1159340 14 2672907076 646725030 53628540 1543500 15 4267591425 965662320 74891040 2016000 16 6616398080 1406064016 102416160 2589440 17 9995653693 2002403718 137494980 3277260 18 14757360516 2796022026 181584900 4093740 19 21343778801 3835983340 236319720 5054000 20 30303773200 5179983060 303519720 6174000 ...
Links
- Max Alekseyev, Derivation of the general formula for U(m,n,k), MathOverflow, 2018.
- Petro Kolosov, On the link between binomial theorem and discrete convolution, arXiv:1603.02468 [math.NT], 2016-2025.
- Petro Kolosov, More details on derivation of present sequence.
- Petro Kolosov, Mathematica program, verifies the identity T^(2m+1) = Sum_{k=0..m} (-1)^(m-k)*U(m,T,k)*T^k for m=0,1,...,12.
- Petro Kolosov, History and overview of the polynomial P_b^m(x), 2024.
- Petro Kolosov, An efficient method of spline approximation for power function, arXiv:2503.07618 [math.GM], 2025.
Crossrefs
Programs
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Mathematica
(* Define the R[n,k] := A302971(m,j)/A304042(m,j) *) 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; (* Define the U(m,l,t) coefficients *) 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}]; (* Define the value of the variable 'm' *) m = 3; (* Print first 10 rows of U(m,l,t) coefficients for 'm' defined above *) Column[Table[U[m, l, t], {l, 1, 10}, {t, 0, m}]]
Formula
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
From Colin Barker, Jul 09 2018; corrected by Max Alekseyev, Sep 06 2018: (Start)
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).
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)
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
Extensions
Edited by Max Alekseyev, Sep 06 2018
Comments