A306409 a(n) = -Sum_{0<=i

0, 1, 3, 10, 34, 120, 434, 1597, 5949, 22363, 84655, 322245, 1232205, 4729453, 18210279, 70307546, 272087770, 1055139408, 4099200524, 15951053566, 62159391150, 242542955378, 947504851414, 3705431067156, 14505084243860, 56831711106496, 222853334131080

Offset: 0

Author

Seiichi Manyama, Apr 05 2019

Keywords

• nonn

Examples

			n | a(n) | A307354 | A006134 | A120305
--+------+---------+---------+---------
0 |    0 |       1 |       1 |       1
1 |    1 |       2 |       3 |       1
2 |    3 |       6 |       9 |       3
3 |   10 |      19 |      29 |       9
4 |   34 |      65 |      99 |      31
5 |  120 |     231 |     351 |     111
		

Links

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

Crossrefs

Partial sums of A014300. - Seiichi Manyama, Jan 30 2023

Cf. A000108, A000957, A006134, A057552, A120305, A307354.

Programs

Mathematica

Table[-Sum[Sum[(-1)^(i+j) * (i+j)!/(i!*j!), {i, 0, j-1}], {j, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Apr 05 2019 *)

PARI

a(n) = -sum(i=0, n, sum(j=i+1, n, (-1)^(i+j)*(i+j)!/(i!*j!)));

PARI

my(N=30, x='x+O('x^N)); concat(0, Vec((1-sqrt(1-4*x))/(sqrt(1-4*x)*(1-x)*(3-sqrt(1-4*x))))) \\ Seiichi Manyama, Jan 30 2023

Formula

a(n) = A006134(n) - A307354(n).

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

a(n) ~ 4^(n+1) / (9*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 05 2019

G.f.: ( 1/(sqrt(1-4*x) * (1-x)) ) * ( x *c(x)/(1 + x *c(x)) ), where c(x) is the g.f. of A000108. - Seiichi Manyama, Jan 30 2023