A320047 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 2-column table read by rows, where n-th row lists coefficients U(1,n,k) for k = 0, 1 and n >= 1.

5, 6, 28, 18, 81, 36, 176, 60, 325, 90, 540, 126, 833, 168, 1216, 216, 1701, 270, 2300, 330, 3025, 396, 3888, 468, 4901, 546, 6076, 630, 7425, 720, 8960, 816, 10693, 918, 12636, 1026, 14801, 1140, 17200, 1260, 19845, 1386, 22748, 1518, 25921, 1656

Offset

1,1

Comments

For l=T, the identity takes the 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

Table of n, a(n) for n=1..46.

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], 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(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.

Conjectures from Colin Barker, Aug 03 2019: (Start)

G.f.: x*(5 + 6*x + 8*x^2 - 6*x^3 - x^4) / ((1 - x)^4*(1 + x)^4).

a(n) = (4 - 4*(-1)^n - 3*(-5+(-1)^n)*n - 3*(-3+(-1)^n)*n^2 + (1+(-1)^(1+n))*n^3) / 8.

a(n) = 4*a(n-2) - 6*a(n-4) + 4*a(n-6) - a(n-8) for n>8.

(End)

Example

        column  column
   l      k=0     k=1
  ---   ------  ------
   1       5       6
   2      28      18
   3      81      36
   4     176      60
   5     325      90
   6     540     126
   7     833     168
   8    1216     216
   9    1701     270
  10    2300     330
  11    3025     396
  12    3888     468
  ...

Mathematica

(* Define the R[n, k] := A302971(n, k)/A304042(n, k) *)
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' to be m = 1 for A320047 *)
m = 1;
(* 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

The case m=2 is A316349.

The case m=3 is A316387.

Column k=0 is A275709.

Column k=1 is A028896.

Cf. A302971, A304042, A287326, A300656, A300785.

Sequence in context: A298175 A298144 A115761 * A249221 A127040 A041011

Adjacent sequences: A320044 A320045 A320046 * A320048 A320049 A320050

Keyword

nonn,tabf

Author

Kolosov Petro, Oct 04 2018

Status

approved