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.
31, 60, 30, 512, 540, 150, 2943, 2160, 420, 10624, 6000, 900, 29375, 13500, 1650, 68256, 26460, 2730, 140287, 47040, 4200, 263168, 77760, 6120, 459999, 121500, 8550, 760000, 181500, 11550, 1199231, 261360, 15180, 1821312, 365040, 19500, 2678143, 496860, 24570, 3830624, 661500, 30450
Offset: 1
Examples
column column column L k=0 k=1 k=2 -- ------- ------- ------ 1 31 60 30 2 512 540 150 3 2943 2160 420 4 10624 6000 900 5 29375 13500 1650 6 68256 26460 2730 7 140287 47040 4200 8 263168 77760 6120 9 459999 121500 8550 10 760000 181500 11550 11 1199231 261360 15180 12 1821312 365040 19500 ...
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', should be m = 2 for A316349 *) m = 2; (* Print first 10 rows of U(m,l,t) coefficients over l: 1 <= l <= 10 *) Column[Table[U[m, l, t], {l, 1, 10}, {t, 0, m}]]
Formula
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
From Colin Barker, Jul 06 2018: (Start)
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).
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)
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.
Extensions
Edited by Max Alekseyev, Sep 06 2018
Comments