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A249939 E.g.f.: 1/(5 - 4*cosh(x)).

%I A249939 #27 Jan 16 2022 08:18:44
%S A249939 1,4,100,6244,727780,136330084,37455423460,14188457293924,
%T A249939 7087539575975140,4514046217675793764,3570250394992512270820,
%U A249939 3433125893070920512725604,3944372161432193963534198500,5336301013125557989981503385444,8396749419933421378024498580446180
%N A249939 E.g.f.: 1/(5 - 4*cosh(x)).
%C A249939 a(n) = 4*A242858(2*n) for n>0.
%C A249939 a(n) = A249940(n)/3.
%C A249939 a(n) == 4 (mod 96) for n>0.
%H A249939 Seiichi Manyama, <a href="/A249939/b249939.txt">Table of n, a(n) for n = 0..200</a>
%F A249939 E.g.f.: 1/3 + (2/3)*Sum_{n>=1} exp(n^2*x) / 2^n  =  Sum_{n>=0} a(n)*x^n/n!.
%F A249939 a(n) = (4/3) * Sum_{k=0..2*n} k! * Stirling2(2*n, k) for n>0 with a(0)=1.
%F A249939 a(n) = Sum_{k=1..[(2*n+1)/3]} 2 * (3*k-1)! * Stirling2(2*n+1, 3*k) for n>0 with a(0)=3, after _Vladimir Kruchinin_ in A242858.
%e A249939 E.g.f.: E(x) = 1 + 4*x^2/2! + 100*x^4/4! + 6244*x^6/6! + 727780*x^8/8! +...
%e A249939 where E(x) = 1/(5 - 4*cosh(x)) = -exp(x) / (2 - 5*exp(x) + 2*exp(2*x)).
%e A249939 ALTERNATE GENERATING FUNCTION.
%e A249939 E.g.f.: A(x) = 1 + 4*x + 100*x^2/2! + 6244*x^3/3! + 727780*x^4/4! +...
%e A249939 where 3*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 + 2*exp(49*x)/2^7 +...
%o A249939 (PARI) /* E.g.f.: 1/(5 - 4*cosh(x)) */
%o A249939 {a(n) = local(X=x+O(x^(2*n+1))); (2*n)!*polcoeff( 1/(5 - 4*cosh(X)), 2*n)}
%o A249939 for(n=0, 20, print1(a(n), ", "))
%o A249939 (PARI) /* Formula for a(n): */
%o A249939 {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
%o A249939 {a(n) = if(n==0, 1, sum(k=1, (2*n+1)\3, 2*(3*k-1)! * Stirling2(2*n+1, 3*k)))}
%o A249939 for(n=0, 20, print1(a(n), ", "))
%o A249939 (PARI) /* Formula for a(n): */
%o A249939 {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
%o A249939 {a(n) = if(n==0, 1, (4/3)*sum(k=0, 2*n, k! * Stirling2(2*n, k) ))}
%o A249939 for(n=0, 20, print1(a(n), ", "))
%Y A249939 Cf. A249938, A249940, A247082, A250914, A250915, A242858.
%Y A249939  Cf. A210676, A210657, A028296, A094088, A210672, A210674.
%K A249939 nonn
%O A249939 0,2
%A A249939 _Paul D. Hanna_, Nov 19 2014