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A247082
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E.g.f.: (8 - 7*cosh(x)) / (13 - 12*cosh(x)).
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5
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1, 5, 365, 66605, 22687565, 12420052205, 9972186170765, 11039636939221805, 16116066766061589965, 29996702068513925975405, 69334618695849722499185165, 194843145588759580915489113005, 654210085817395711127396030796365, 2586566313303319454399746941903834605, 11894287668430209899882926599828701863565
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OFFSET
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0,2
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COMMENTS
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The number of 3-level labeled linear rooted trees with 2*n leaves.
a(n) == 5 (mod 360) for n>0.
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LINKS
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FORMULA
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E.g.f.: 1/2 + (1/6)*Sum_{n>=0} exp(n^2*x) * (2/3)^n = Sum_{n>=0} a(n)*x^n/n!.
a(n) = Sum_{k=0..2*n} 2^(k-1) * k! * Stirling2(2*n, k) for n>0 with a(0)=1. [After Vladeta Jovovic in A050351]
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EXAMPLE
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E.g.f.: E(x) = 1 + 5*x^2/2! + 365*x^4/4! + 66605*x^6/6! + 22687565*x^8/8! +...
where E(x) = (8 - 7*cosh(x)) / (13 - 12*cosh(x)), or, equivalently,
E(x) = (7 - 16*exp(x) + 7*exp(2*x)) / (12 - 26*exp(x) + 12*exp(2*x)).
ALTERNATE GENERATING FUNCTION.
E.g.f.: A(x) = 1 + 5*x + 365*x^2/2! + 66605*x^3/3! + 22687565*x^4/4! +...
where
6*A(x) = 4 + exp(x)*(2/3) + exp(4*x)*(2/3)^2 + exp(9*x)*(2/3)^3 + exp(16*x)*(2/3)^4 + exp(25*x)*(2/3)^5 + exp(36*x)*(2/3)^6 + exp(49*x)*(2/3)^7 +...
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MATHEMATICA
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nmax=20; Table[(CoefficientList[Series[(8-7*Cosh[x]) / (13-12*Cosh[x]), {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[n]], {n, 1, 2*nmax+2, 2}] (* Vaclav Kotesovec, Nov 29 2014 *)
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PROG
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(PARI) /* E.g.f.: (8 - 7*cosh(x)) / (13 - 12*cosh(x)): */
{a(n) = local(X=x+O(x^(2*n+1))); (2*n)!*polcoeff( (8 - 7*cosh(X)) / (13 - 12*cosh(X)) , 2*n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* Formula for a(n): */
{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = if(n==0, 1, sum(k=0, 2*n, 2^(k-1) * k! * Stirling2(2*n, k) ))}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* As the Sum of an Infinite Series: */
\p60 \\ set precision
Vec(serlaplace(1/2+1/6*sum(n=0, 2000, exp(n^2*x)*(2/3)^n*1.)))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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