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A249940 E.g.f.: 1 + Sum_{n>=1} 2*exp(n^2*x) / 2^n.

%I A249940 #16 Nov 29 2014 14:54:43
%S A249940 3,12,300,18732,2183340,408990252,112366270380,42565371881772,
%T A249940 21262618727925420,13542138653027381292,10710751184977536812460,
%U A249940 10299377679212761538176812,11833116484296581890602595500,16008903039376673969944510156332,25190248259800264134073495741338540
%N A249940 E.g.f.: 1 + Sum_{n>=1} 2*exp(n^2*x) / 2^n.
%F A249940 E.g.f.: 3/(5 - 4*cosh(x)) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!.
%F A249940 a(n) = Sum_{k=0..2*n} 4*k! * Stirling2(2*n, k) for n>0 with a(0)=3.
%F A249940 a(n) = Sum_{k=1..[(2*n+1)/3]} 2*(3*k)! * Stirling2(2*n+1,3*k) / k for n>0 with a(0)=3, after _Vladimir Kruchinin_ in A242858.
%F A249940 a(n) ~ 2 * (2*n)! / (log(2))^(2*n+1). - _Vaclav Kotesovec_, Nov 29 2014
%e A249940 E.g.f.: A(x) = 3 + 12*x + 300*x^2/2! + 18732*x^3/3! + 2183340*x^4/4! +...
%e A249940 where the e.g.f. equals the infinite series:
%e A249940 A(x) = 1 + 2*exp(x)/2 + 2*exp(4*x)/2^2 + 2*exp(9*x)/2^3 + 2*exp(16*x)/2^4 + 2*exp(25*x)/2^5 + 2*exp(36*x)/2^6 +...
%e A249940 We also have the following series expansion:
%e A249940 3/(5 - 4*cosh(x)) = 3 + 12*x^2/2! + 300*x^4/4! + 18732*x^6/6! + 2183340*x^8/8! + 408990252*x^10/10! +...
%t A249940 nmax=20; Table[(CoefficientList[Series[3/(5-4*Cosh[x]), {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[n]],{n,1,2*nmax+2,2}] (* _Vaclav Kotesovec_, Nov 29 2014 *)
%o A249940 (PARI) /* E.g.f.: 3/(5 - 4*cosh(x)): */
%o A249940 {a(n) = local(X=x+O(x^(2*n+1))); (2*n)!*polcoeff( 3/(5 - 4*cosh(X)), 2*n)}
%o A249940 for(n=0,20,print1(a(n),", "))
%o A249940 (PARI) /* E.g.f.: 1 + Sum_{n>=1} 2*exp(n^2*x)/2^n */
%o A249940 \p100 \\ set precision
%o A249940 {a(n) = round( n!*polcoeff(1+2*sum(m=1,500,exp(m^2*x +x*O(x^n))/2^m*1.), n))}
%o A249940 for(n=0,20,print1(a(n),", "))
%o A249940 (PARI) /* Formula for a(n): */
%o A249940 {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
%o A249940 {a(n) = if(n==0,3, sum(k=0, 2*n, 4*k! * Stirling2(2*n, k) ))}
%o A249940 for(n=0, 20, print1(a(n), ", "))
%o A249940 (PARI) /* Formula for a(n): */
%o A249940 {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
%o A249940 {a(n) = if(n==0,3, 2*sum(k=1,(2*n+1)\3, (3*k)! * Stirling2(2*n+1,3*k) / k))}
%o A249940 for(n=0,20,print1(a(n),", "))
%Y A249940 Cf. A242858, A249941.
%K A249940 nonn
%O A249940 0,1
%A A249940 _Paul D. Hanna_, Nov 19 2014