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A307354
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a(n) = Sum_{0<=i<=j<=n} (-1)^(i+j) * (i+j)!/(i!*j!).
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5
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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
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OFFSET
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0,2
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LINKS
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FORMULA
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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)
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)
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MATHEMATICA
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Table[Sum[Sum[(-1)^(i + j)*(i + j)!/(i!*j!), {i, 0, j}], {j, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Apr 04 2019 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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