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%I #25 Oct 23 2023 12:41:03
%S 1,5,49,709,13505,318181,8916145,289283429,10656031489,439039941445,
%T 19995858681521,997184081617285,54026137182982849,3159127731435043109,
%U 198258247783634075185,13289190424904891606821,947419111092028780186625
%N a(n) = Sum_{i=0..n} binomial(n,i)^2*i!*4^i.
%H Seiichi Manyama, <a href="/A102773/b102773.txt">Table of n, a(n) for n = 0..363</a>
%H Z. Li, Z. Li and Y. Cao, <a href="https://doi.org/10.1016/j.disc.2006.03.047">Enumeration of symplectic and orthogonal injective partial transformations</a>, Discrete Math., 306 (2006), 1781-1787. (The function s_n.)
%F E.g.f.: (1/(1-4x))*exp(x/(1-4x)).
%F a(n) = (8*n-3)*a(n-1) - 16*(n-1)^2*a(n-2). - _Vaclav Kotesovec_, Oct 09 2013
%F a(n) ~ n^(n+1/4) * exp(sqrt(n)-n-1/8) * 4^n * (1 + 37/(96*sqrt(n))). - _Vaclav Kotesovec_, Oct 09 2013
%F Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(4*x) * BesselI(0,2*sqrt(x)). - _Ilya Gutkovskiy_, Jul 17 2020
%p seq(sum('binomial(k,i)^2*i!*4^i', 'i'=0..k),k=0..30);
%t f[n_] := Sum[k!*4^k*Binomial[n, k]^2, {k, 0, n}]; Table[ f[n], {n, 0, 16}] (* or *)
%t Range[0, 16]! CoefficientList[ Series[1/(1 - 4x)*Exp[x/(1 - 4x)], {x, 0, 16}], x] (* _Robert G. Wilson v_, Mar 16 2005 *)
%o (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
%Y Cf. A002720, A025167, A289147.
%K easy,nonn
%O 0,2
%A _Miklos Kristof_, Mar 16 2005
%E More terms from _Robert G. Wilson v_, Mar 16 2005
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