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%I #54 Jan 30 2023 09:24:52
%S 0,1,3,10,34,120,434,1597,5949,22363,84655,322245,1232205,4729453,
%T 18210279,70307546,272087770,1055139408,4099200524,15951053566,
%U 62159391150,242542955378,947504851414,3705431067156,14505084243860,56831711106496,222853334131080
%N a(n) = -Sum_{0<=i<j<=n} (-1)^(i+j) * (i+j)!/(i!*j!).
%H Seiichi Manyama, <a href="/A306409/b306409.txt">Table of n, a(n) for n = 0..1664</a>
%F a(n) = A006134(n) - A307354(n).
%F a(n) = (A006134(n) - A120305(n))/2.
%F a(n) ~ 4^(n+1) / (9*sqrt(Pi*n)). - _Vaclav Kotesovec_, Apr 05 2019
%F 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
%e n | a(n) | A307354 | A006134 | A120305
%e --+------+---------+---------+---------
%e 0 | 0 | 1 | 1 | 1
%e 1 | 1 | 2 | 3 | 1
%e 2 | 3 | 6 | 9 | 3
%e 3 | 10 | 19 | 29 | 9
%e 4 | 34 | 65 | 99 | 31
%e 5 | 120 | 231 | 351 | 111
%t 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 *)
%o (PARI) a(n) = -sum(i=0, n, sum(j=i+1, n, (-1)^(i+j)*(i+j)!/(i!*j!)));
%o (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
%Y Partial sums of A014300. - _Seiichi Manyama_, Jan 30 2023
%Y Cf. A000108, A000957, A006134, A057552, A120305, A307354.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Apr 05 2019
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