A307354 a(n) = Sum_{0<=i<=j<=n} (-1)^(i+j) * (i+j)!/(i!*j!).

1, 2, 6, 19, 65, 231, 841, 3110, 11628, 43834, 166298, 634140, 2428336, 9331688, 35967462, 138987715, 538287881, 2088842463, 8119916647, 31613327405, 123251518641, 481125828853, 1880262896537, 7355767408395, 28803717914791, 112887697489907, 442784607413427

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1664

Formula

a(n) = (A006134(n) + A120305(n))/2.

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

Partial sums of A026641. - Seiichi Manyama, Jan 30 2023

Cf. A000108, A006134, A079309, A105872, A120305, A306409.

Cf. A191993, A360186, A360212.

Sequence in context: A220065 A348848 A047016 * A307487 A212380 A150084

Adjacent sequences: A307351 A307352 A307353 * A307355 A307356 A307357

Keyword

nonn

Author

Seiichi Manyama, Apr 03 2019

Status

approved