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

%I A307354 #36 Apr 08 2024 09:12:52
%S A307354 1,2,6,19,65,231,841,3110,11628,43834,166298,634140,2428336,9331688,
%T A307354 35967462,138987715,538287881,2088842463,8119916647,31613327405,
%U A307354 123251518641,481125828853,1880262896537,7355767408395,28803717914791,112887697489907,442784607413427
%N A307354 a(n) = Sum_{0<=i<=j<=n} (-1)^(i+j) * (i+j)!/(i!*j!).
%H A307354 Seiichi Manyama, <a href="/A307354/b307354.txt">Table of n, a(n) for n = 0..1664</a>
%F A307354 a(n) = (A006134(n) + A120305(n))/2.
%F A307354 From _Vaclav Kotesovec_, Apr 04 2019: (Start)
%F A307354 Recurrence: 2*n*a(n) = (9*n-4)*a(n-1) - (3*n-2)*a(n-2) - 2*(2*n-1)*a(n-3).
%F A307354 a(n) ~ 2^(2*n+3) / (9*sqrt(Pi*n)). (End)
%F A307354 From _Seiichi Manyama_, Jan 29 2023: (Start)
%F A307354 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2*n-3*k,n).
%F A307354 G.f.: 1 / ( sqrt(1-4*x) * (1 + x^3 * c(x)^3) ), where c(x) is the g.f. of A000108. (End)
%F A307354 a(n) = [x^n] 1/((1+x^3) * (1-x)^(n+1)). - _Seiichi Manyama_, Apr 08 2024
%t A307354 Table[Sum[Sum[(-1)^(i + j)*(i + j)!/(i!*j!), {i, 0, j}], {j, 0, n}], {n, 0, 25}] (* _Vaclav Kotesovec_, Apr 04 2019 *)
%o A307354 (PARI) a(n) = sum(i=0, n, sum(j=i, n, (-1)^(i+j)*(i+j)!/(i!*j!)));
%o A307354 (PARI) a(n) = sum(k=0, n\3, (-1)^k*binomial(2*n-3*k, n)); \\ _Seiichi Manyama_, Jan 29 2023
%o A307354 (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
%Y A307354 Partial sums of A026641. - _Seiichi Manyama_, Jan 30 2023
%Y A307354 Cf. A000108, A006134, A079309, A105872, A120305, A306409.
%Y A307354 Cf. A191993, A360186, A360212.
%K A307354 nonn
%O A307354 0,2
%A A307354 _Seiichi Manyama_, Apr 03 2019