A325517 a(n) = n*((2*n + 1)*(2*n^2 + 2*n + 3) - 3*(-1)^n)/24.

0, 1, 6, 24, 64, 145, 282, 504, 832, 1305, 1950, 2816, 3936, 5369, 7154, 9360, 12032, 15249, 19062, 23560, 28800, 34881, 41866, 49864, 58944, 69225, 80782, 93744, 108192, 124265, 142050, 161696, 183296, 207009, 232934, 261240, 292032, 325489, 361722, 400920, 443200

Offset: 0

Author

Stefano Spezia, May 07 2019

Keywords

• nonn
• easy

Comments

For n > 0, a(n) is the n-th row sum of the triangle A325516.

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. A131941, A317614, A322277, A325516.

Programs

GAP

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

Magma

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

Maple

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

Mathematica

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

PARI

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

Formula

O.g.f.: x*(1 + 3*x + 7*x^2 + 3*x^3 + 2*x^4)/((1 - x)^5*(1 + x)^2).

E.g.f.: (1/24)*exp(-x)*x*(3 + 21*exp(2*x) + 54*exp(2*x)*x + 30*exp(2*x)*x^2 + 4*exp(2*x)*x^3).

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

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

a(n) = n*A131941(n). - Stefano Spezia, Dec 21 2021