OFFSET
0,2
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1664
FORMULA
From Vaclav Kotesovec, Apr 04 2019: (Start)
Recurrence: 2*n*a(n) = (9*n-4)*a(n-1) - (3*n-2)*a(n-2) - 2*(2*n-1)*a(n-3).
a(n) ~ 2^(2*n+3) / (9*sqrt(Pi*n)). (End)
From Seiichi Manyama, Jan 29 2023: (Start)
a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2*n-3*k,n).
G.f.: 1 / ( sqrt(1-4*x) * (1 + x^3 * c(x)^3) ), where c(x) is the g.f. of A000108. (End)
a(n) = [x^n] 1/((1+x^3) * (1-x)^(n+1)). - Seiichi Manyama, Apr 08 2024
MATHEMATICA
Table[Sum[Sum[(-1)^(i + j)*(i + j)!/(i!*j!), {i, 0, j}], {j, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Apr 04 2019 *)
PROG
(PARI) a(n) = sum(i=0, n, sum(j=i, n, (-1)^(i+j)*(i+j)!/(i!*j!)));
(PARI) a(n) = sum(k=0, n\3, (-1)^k*binomial(2*n-3*k, n)); \\ Seiichi Manyama, Jan 29 2023
(PARI) my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1+x^3*(2/(1+sqrt(1-4*x)))^3))) \\ Seiichi Manyama, Jan 29 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Apr 03 2019
STATUS
approved