A159960 a(n) = (1/2) * Sum_{k=1..n} (-1)^(k-1) * binomial(2*n-k, k) * binomial(n, k) * 2^k * (2*n-2*k)!.
1, 10, 292, 16152, 1443616, 189709600, 34420171584, 8241995095936, 2517637537094656, 955377719901439488, 440888939541736115200, 243144648530111594371072, 157920570527279020394569728, 119308432982412667510831095808, 103738687936577909824307104989184
Offset: 1
Links
- G. C. Greubel, Table of n, a(n) for n = 1..220
Programs
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Maple
f := proc (n) add((-1)^(k-1)*binomial(2*n-k, k)*binomial(n, k)*2^k*factorial(2*n-2*k), k = 1 .. n)/2 end proc;
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Mathematica
a[n_] := (2*n)!*(1-HypergeometricPFQ[{-n}, {1, -2*n}, -2])/2; Table[a[n], {n, 1, 15}] (* Jean-François Alcover, Jan 27 2014 *) -
PARI
a(n)=sum(k=1,n,(-1)^(k-1)*binomial(2*n-k,k)*binomial(n, k)<
Formula
a(n) = (1/2) * Sum_{k=1..n} (-1)^(k-1) * binomial(2*n-k, k) * binomial(n, k) * 2^k * (2*n-2*k)!.
Recurrence: (6*n - 17)*a(n) = 2*(n-1)*(36*n^2 - 156*n + 151)*a(n-1) - 4*(n-1)*(72*n^4 - 636*n^3 + 2062*n^2 - 2909*n + 1511)*a(n-2) + 4*(n-2)*(n-1)*(96*n^5 - 1280*n^4 + 6704*n^3 - 17208*n^2 + 21596*n - 10569)*a(n-3) + 8*(n-3)*(n-2)*(n-1)*(2*n - 7)*(6*n - 11)*a(n-4). - Vaclav Kotesovec, Mar 15 2014
a(n) ~ (1-BesselJ(0,2)) * sqrt(Pi) * 4^n * n^(2*n+1/2) / exp(2*n). - Vaclav Kotesovec, Mar 15 2014
Extensions
Name corrected and edited by Giovanni Resta, Aug 12 2025
Comments