A338634 G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 2^4*x/(A(x) - 3^4*x/(A(x) - 4^4*x/(A(x) - 5^4*x/(A(x) - 6^4*x/(A(x) - ... )))))), a continued fraction relation.
1, 1, 15, 1490, 472475, 367254494, 596838469302, 1812465211795364, 9460229930323620755, 79588323526110945959270, 1025816228173896271039326050, 19441688693651416990291991566332, 523762848713992063145153491388390686, 19495503038639783268900576813041922912172
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 15*x^2 + 1490*x^3 + 472475*x^4 + 367254494*x^5 + 596838469302*x^6 + 1812465211795364*x^7 + 9460229930323620755*x^8 + ... where 1 = A(x) - x/(A(x) - 2^4*x/(A(x) - 3^4*x/(A(x) - 4^4*x/(A(x) - 5^4*x/(A(x) - 6^4*x/(A(x) - 7^4*x/(A(x) - 8^4*x/(A(x) - 9^4*x/(A(x) - 10^4*x/(A(x) - ... )))))))))), a continued fraction relation.
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..130
Programs
-
PARI
{a(n) = my(A=[1],CF=1); for(i=1,n, A=concat(A,0); for(i=1,#A, CF = Ser(A) - (#A-i+1)^4*x/CF ); A[#A] = -polcoeff(CF,#A-1) );A[n+1] } for(n=0,20,print1(a(n),", "))
Formula
For n > 0, a(n) is odd iff n is a power of 2 (conjecture).
From Vaclav Kotesovec, Nov 12 2020: (Start)
a(n) ~ sqrt(2/Pi) * (8*sqrt(Pi) / Gamma(1/4)^2)^(4*n + 2) * (n!)^4 / sqrt(n).
a(n) ~ 2^(12*n + 17/2) * Pi^(2*n + 5/2) * n^(4*n + 3/2) / (Gamma(1/4)^(8*n + 4) * exp(4*n)). (End)