A325656 a(n) = (1/24)*n*((4*n + 3)*(2*n^2 + 1) - 3*(-1)^n).

0, 1, 8, 36, 104, 245, 492, 896, 1504, 2385, 3600, 5236, 7368, 10101, 13524, 17760, 22912, 29121, 36504, 45220, 55400, 67221, 80828, 96416, 114144, 134225, 156832, 182196, 210504, 242005, 276900, 315456, 357888, 404481, 455464, 511140, 571752, 637621, 709004, 786240

Offset: 0

Author

Stefano Spezia, May 13 2019

Keywords

• nonn
• easy

Comments

For n > 0, a(n) is the n-th row sum of the triangle A325655.

Links

• Stefano Spezia, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Crossrefs

Cf. A173722, A317614, A322277, A325655.

Programs

GAP

Flat(List([0..50], n->(1/24)*n*(3 - 3*(- 1)^n + 4*n + 6*n^2 + 8*n^3)));

Magma

[(1/24)*n*(3 - 3*(- 1)^n + 4*n + 6*n^2 + 8*n^3): n in [0..50]];

Maple

a:=n->(1/24)*n*(3 - 3*(- 1)^n + 4*n + 6*n^2 + 8*n^3): seq(a(n), n=0..50);

Mathematica

a[n_]:=(1/24)*n*(3 - 3*(- 1)^n + 4*n + 6*n^2 + 8*n^3); Array[a,50,0]

PARI

a(n) = (1/24)*n*(3 - 3*(- 1)^n + 4*n + 6*n^2 + 8*n^3);

Formula

O.g.f.: -x*(1 + 5*x + 13*x*2 + 9*x^3 + 4*x^4)/((-1 + x)^5*(1 + x)^2).

E.g.f.: (1/24)*exp(-x)*x*(3 + 21*exp(2*x) + 78*exp(2*x)*x + 54*exp(2*x)*x^2 + 8*exp(2*x)*x*3).

a(n) = a(n) = 3*a(n-1) - a(n-2) - 5*a(n-3) + 5*a(n-4) + a(n-5) - 3*a(n-6) + a(n-7) for n > 6.

a(n) = (1/12)*n^2*(4*n^2 + 3*n + 2) if n is even.

a(n) = (1/12)*n*(n + 1)*(4*n^2 - n + 3) if n is odd.

a(n) = n*A173722(2*n). - Stefano Spezia, Dec 21 2021