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

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