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A250914
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E.g.f.: (18 - 17*cosh(x)) / (25 - 24*cosh(x)).
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5
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1, 7, 1015, 367927, 248956855, 270732878647, 431806658432695, 949587798053709367, 2753726282896986372535, 10181613308681289633868087, 46749244630988859672950920375, 260970234691672017384493753162807, 1740621952318191255997909826897420215, 13670746044282245244660044262911331401527
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OFFSET
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0,2
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COMMENTS
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The number of 4-level labeled linear rooted trees with 2*n leaves.
a(n) == 7 (mod 1008) for n>0.
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LINKS
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FORMULA
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E.g.f.: 2/3 + (1/12)*Sum_{n>=0} exp(n^2*x) * (3/4)^n = Sum_{n>=0} a(n)*x^n/n!.
a(n) = Sum_{k=0..2*n} 3^(k-1) * k! * Stirling2(2*n, k) for n>0 with a(0)=1.
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EXAMPLE
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E.g.f.: E(x) = 1 + 7*x^2/2! + 1015*x^4/4! + 367927*x^6/6! + 248956855*x^8/8! +...
where E(x) = (18 - 17*cosh(x)) / (25 - 24*cosh(x)).
ALTERNATE GENERATING FUNCTION.
E.g.f.: A(x) = 1 + 7*x + 1015*x^2/2! + 367927*x^3/3! + 248956855*x^4/4! +...
where
12*A(x) = 9 + exp(x)*(3/4) + exp(4*x)*(3/4)^2 + exp(9*x)*(3/4)^3 + exp(16*x)*(3/4)^4 + exp(25*x)*(3/4)^5 + exp(36*x)*(3/4)^6 +...
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MATHEMATICA
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nmax=20; Table[(CoefficientList[Series[(18-17*Cosh[x]) / (25-24*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.: (18 - 17*cosh(x)) / (25 - 24*cosh(x)): */
{a(n) = local(X=x+O(x^(2*n+1))); (2*n)!*polcoeff( (18 - 17*cosh(X)) / (25 - 24*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, 3^(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(2/3 + 1/12*sum(n=0, 2000, exp(n^2*x)*(3/4)^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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