OFFSET
1,1
COMMENTS
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.
LINKS
Max Alekseyev, Derivation of the general formula for U(m,n,k), MathOverflow, 2018.
Petro Kolosov, Series Representation of Power Function, arXiv:1603.02468 [math.NT], 2016-2018.
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.
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.
EXAMPLE
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
...
MATHEMATICA
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}]]
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Kolosov Petro, Jun 29 2018
EXTENSIONS
Edited by Max Alekseyev, Sep 06 2018
STATUS
approved