OFFSET
1,1
COMMENTS
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.
LINKS
Max Alekseyev, Derivation of the general formula for U(m,n,k), MathOverflow, 2018.
Petro Kolosov, Another Power Identity involving Binomial Theorem and Faulhaber's formula, arXiv:1603.02468 [math.NT], 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(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
EXAMPLE
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
...
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' *)
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}]]
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Kolosov Petro, Jul 01 2018
EXTENSIONS
Edited by Max Alekseyev, Sep 06 2018
STATUS
approved