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A099022
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a(n) = Sum_{k=0..n} C(n,k)*(2*n-k)!.
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6
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1, 3, 38, 1158, 65304, 5900520, 780827760, 142358474160, 34209760152960, 10478436416945280, 3984884716852972800, 1842169367191937414400, 1017403495472574045158400, 661599650478455071589606400, 500354503197888042597961267200, 435447353708763072625260119808000
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OFFSET
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0,2
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COMMENTS
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Diagonal of Euler-Seidel matrix with start sequence n!.
Number of ways to use the elements of {1,..,k}, n<=k<=2n, once each to form a sequence of n lists, each having length 1 or 2. - Bob Proctor, Apr 18 2005, Jun 26 2006
Replace "lists" by "sets": A105749.
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LINKS
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FORMULA
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T(2*n, n), where T is the triangle in A076571.
A082765(n) = Sum[C(n, k)*a(k), 0<=k<=n].
a(n) = int {x = 0..inf} exp(-x)*(x + x^2)^n dx. Applying the results of Nicolaescu, Section 3.2 to this integral we obtain the asymptotic expansion a(n) ~ (2*n)!*exp(1/2)*( 1 - 1/(16*n) - 191/(6144*n^2) + O(1/n^3) ). - Peter Bala, Jul 07 2014
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MAPLE
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f:= gfun:-rectoproc({a(n)=2*n*(2*n-1)*a(n-1)+n*(n-1)*a(n-2), a(0)=1, a(1)=3}, a(n), remember):
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MATHEMATICA
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Table[Sum[Binomial[n, k](2n-k)!, {k, 0, n}], {n, 0, 20}] (* Harvey P. Dale, Nov 22 2021 *)
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PROG
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(PARI) for(n=0, 25, print1(sum(k=0, n, binomial(n, k)*(2*n-k)!), ", ")) \\ G. C. Greubel, Dec 31 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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