A102773 a(n) = Sum_{i=0..n} binomial(n,i)^2*i!*4^i.
1, 5, 49, 709, 13505, 318181, 8916145, 289283429, 10656031489, 439039941445, 19995858681521, 997184081617285, 54026137182982849, 3159127731435043109, 198258247783634075185, 13289190424904891606821, 947419111092028780186625
Offset: 0
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..363
- Z. Li, Z. Li and Y. Cao, Enumeration of symplectic and orthogonal injective partial transformations, Discrete Math., 306 (2006), 1781-1787. (The function s_n.)
Programs
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Maple
seq(sum('binomial(k,i)^2*i!*4^i', 'i'=0..k),k=0..30); -
Mathematica
f[n_] := Sum[k!*4^k*Binomial[n, k]^2, {k, 0, n}]; Table[ f[n], {n, 0, 16}] (* or *) Range[0, 16]! CoefficientList[ Series[1/(1 - 4x)*Exp[x/(1 - 4x)], {x, 0, 16}], x] (* Robert G. Wilson v, Mar 16 2005 *) -
PARI
a(n)=my(t=1); sum(i=1,n, t*=i; binomial(n,i)^2*t<<(2*i), 1) \\ Charles R Greathouse IV, Oct 23 2023
Formula
E.g.f.: (1/(1-4x))*exp(x/(1-4x)).
a(n) = (8*n-3)*a(n-1) - 16*(n-1)^2*a(n-2). - Vaclav Kotesovec, Oct 09 2013
a(n) ~ n^(n+1/4) * exp(sqrt(n)-n-1/8) * 4^n * (1 + 37/(96*sqrt(n))). - Vaclav Kotesovec, Oct 09 2013
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(4*x) * BesselI(0,2*sqrt(x)). - Ilya Gutkovskiy, Jul 17 2020
Extensions
More terms from Robert G. Wilson v, Mar 16 2005