A254030 a(n) = 1*4^n + 2*3^n + 3*2^n + 4*1^n.

10, 20, 50, 146, 470, 1610, 5750, 21146, 79430, 303050, 1169750, 4554746, 17852390, 70322090, 278050550, 1102537946, 4381257350, 17438542730, 69495104150, 277204002746, 1106488342310, 4418973508970, 17654960746550

Offset: 0

Author

Luciano Ancora, Jan 26 2015

Keywords

• nonn
• easy

Comments

This is the sequence of fourth 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 (10,-35,50,-24).

Crossrefs

Cf. A052548, A254028, A254031, A254144, A254145, A254146.

Programs

Maple

seq(add(i*(5 - i)^n, i = 1..4), n = 0..20); # Peter Bala, Jan 31 2017

Mathematica

Table[3 2^n + 2^(2 n) + 2 3^n + 4, {n, 0, 25}] (* Bruno Berselli, Jan 27 2015 *)
LinearRecurrence[{10,-35,50,-24},{10,20,50,146},30] (* Harvey P. Dale, Jun 06 2020 *)

PARI

Vec(-2*(77*x^3-100*x^2+40*x-5)/((x-1)*(2*x-1)*(3*x-1)*(4*x-1))  + O(x^100)) \\ Colin Barker, Jan 26 2015

Formula

G.f.: -2*(77*x^3-100*x^2+40*x-5) / ((x-1)*(2*x-1)*(3*x-1)*(4*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 = 4.

a(n) = 1/3!*Sum_{k = 0..n} (-1)^(k+n)*(k + 5)!*Stirling2(n,k)/

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

E.g.f.: exp(x)*(4 + 3*exp(x) + 2*exp(2*x) + exp(3*x)). - Stefano Spezia, May 19 2025