login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

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.
3

%I #9 Nov 12 2020 12:00:25

%S 1,1,15,1490,472475,367254494,596838469302,1812465211795364,

%T 9460229930323620755,79588323526110945959270,

%U 1025816228173896271039326050,19441688693651416990291991566332,523762848713992063145153491388390686,19495503038639783268900576813041922912172

%N 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.

%H Paul D. Hanna, <a href="/A338634/b338634.txt">Table of n, a(n) for n = 0..130</a>

%F For n > 0, a(n) is odd iff n is a power of 2 (conjecture).

%F From _Vaclav Kotesovec_, Nov 12 2020: (Start)

%F a(n) ~ sqrt(2/Pi) * (8*sqrt(Pi) / Gamma(1/4)^2)^(4*n + 2) * (n!)^4 / sqrt(n).

%F 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)

%e 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 + ...

%e where

%e 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.

%o (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] }

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A000699, A158119, A227887, A338633, A338635.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 04 2020