A245154 E.g.f.: (cosh(3*x) + sinh(3*x)*cosh(x)) / sqrt(1 - sinh(x)^2*sinh(3*x)^2).
1, 3, 9, 36, 189, 2148, 26109, 371136, 5407929, 95795568, 1832049009, 41428038336, 972380766069, 25736128903488, 705111069908709, 21600790506395136, 683861855417706609, 23836956839153265408, 853476673589938069209, 33263825890074489025536
Offset: 0
Keywords
Examples
E.g.f.: A(x) = 1 + 3*x + 9*x^2/2! + 36*x^3/3! + 189*x^4/4! + 2148*x^5/5! +... Let A(x) = A0(x) + A1(x) where A0(x) = 1 + 9*x^2/2! + 189*x^4/4! + 26109*x^6/6! + 5407929*x^8/8! +... A1(x) = 3*x + 36*x^3/3! + 2148*x^5/5! + 371136*x^7/7! + 95795568*x^9/9! +... then A0(x)^2 - A1(x)^2 = 1. Note that the logarithm of the e.g.f. is an odd function: Log(A(x)) = 3*x + 9*x^3/3! + 1095*x^5/5! + 119469*x^7/7! + 28399275*x^9/9! + 11494484529*x^11/11! + 6432743099055*x^13/13! +... thus A(x)*A(-x) = 1.
Programs
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PARI
{a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff((cosh(3*X) + sinh(3*X)*cosh(X)) / sqrt(1 - sinh(X)^2*sinh(3*X)^2), n)} for(n=0, 30, print1(a(n), ", "))
Formula
E.g.f.: G(x) * (cosh(x) - sinh(x)*cosh(3*x)) / sqrt(1 - sinh(x)^2*sinh(3*x)^2), where G(x) is the e.g.f. of A245155.
a(n) ~ 2*sqrt(2) * n^n / (5^(1/4) * exp(n) * (log((1+sqrt(5))/2))^(n+1/2)). - Vaclav Kotesovec, Nov 04 2014
Comments