A254031 a(n) = 1*5^n + 2*4^n + 3*3^n + 4*2^n + 5*1^n.

15, 35, 105, 371, 1449, 6035, 26265, 117971, 542409, 2538515, 12044025, 57756371, 279305769, 1359736595, 6654800985, 32708239571, 161307227529, 797687136275, 3953299529145, 19626731023571, 97576919443689, 485664640673555

Offset: 0

Author

Luciano Ancora, Jan 26 2015

Keywords

• nonn
• easy

Comments

This is the sequence of fifth terms of "second partial sums of m-th powers".

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Luciano Ancora, Demonstration of formulas
• Index entries for linear recurrences with constant coefficients, signature (15,-85,225,-274,120).

Crossrefs

Cf. A052548, A254028, A254030, A254144, A254145, A254146.

Programs

Maple

seq(add(i*(6 - i)^n, i = 1..5), n = 0..20); # Peter Bala, Jan 31 2017

Mathematica

Table[2^(n + 2) + 2^(2 n + 1) + 3^(n + 1) + 5^n + 5, {n, 0, 25}] (* Bruno Berselli, Jan 27 2015 *)
LinearRecurrence[{15,-85,225,-274,120},{15,35,105,371,1449},30] (* Harvey P. Dale, Jan 24 2022 *)

PARI

Vec(-(1044*x^4-1604*x^3+855*x^2-190*x+15)/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)) + O(x^100)) \\ Colin Barker, Jan 26 2015

Formula

G.f.: -(1044*x^4 - 1604*x^3 + 855*x^2 - 190*x + 15) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)). - Colin Barker, Jan 26 2015

From Peter Bala, Jan 31 2016: (Start)

a(n) = (x + 1)*( Bernoulli(n + 1, x + 1) - Bernoulli(n + 1, 1) )/(n + 1) - ( Bernoulli(n + 2, x + 1) - Bernoulli(n + 2, 1) )/(n + 2) at x = 5.

a(n) = (1/4!)*Sum_{k = 0..n} (-1)^(k+n)*(k + 6)!*Stirling2(n,k)/

((k + 1)*(k + 2)). (End)