A338635 - OEIS

%I #14 Nov 12 2020 12:10:46

%S 1,1,2,17,274,6749,231276,10465440,604220826,43388420549,

%T 3797054582794,398157728106929,49311011342018168,7124133759620985652,

%U 1187818792835133749984,226420783437860189825400,48936975180367428260159850,11904986360488865549641429797,3238569202146221391019821488694

%N G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 3*x/(A(x) - 6*x/(A(x) - 10*x/(A(x) - 15*x/(A(x) - 21*x/(A(x) - 28*x/(A(x) - ... - (n*(n+1)/2)*x/(A(x) - ...))))))), a continued fraction relation.

%H Paul D. Hanna, <a href="/A338635/b338635.txt">Table of n, a(n) for n = 0..200</a>

%F For n>0, a(n) is odd iff n = A118113(k) for some k >= 1, where A118113(k) = 2*Fibbinary(k) + 1 (conjecture).

%F a(n) ~ 2^(3*n + 5) * n^(2*n + 3/2) / (Pi^(2*n + 3/2) * exp(2*n)). - _Vaclav Kotesovec_, Nov 12 2020

%e G.f.: A(x) = 1 + x + 2*x^2 + 17*x^3 + 274*x^4 + 6749*x^5 + 231276*x^6 + 10465440*x^7 + 604220826*x^8 + 43388420549*x^9 + 3797054582794*x^10 + ...

%e where

%e 1 = A(x) - x/(A(x) - 3*x/(A(x) - 6*x/(A(x) - 10*x/(A(x) - 15*x/(A(x) - 21*x/(A(x) - 28*x/(A(x) - 36*x/(A(x) - 45*x/(A(x) - 55*x/(A(x) - ...)))))))))), a continued fraction relation in which the triangular numbers appear as coefficients.

%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)*(#A-i+2)/2*x/CF ); A[#A] = -polcoeff(CF,#A-1) );A[n+1] }

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

%Y Cf. A000699, A158119, A338633, A338634, A118113.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 04 2020