A338635 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.

1, 1, 2, 17, 274, 6749, 231276, 10465440, 604220826, 43388420549, 3797054582794, 398157728106929, 49311011342018168, 7124133759620985652, 1187818792835133749984, 226420783437860189825400, 48936975180367428260159850, 11904986360488865549641429797, 3238569202146221391019821488694

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

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

a(n) ~ 2^(3*n + 5) * n^(2*n + 3/2) / (Pi^(2*n + 3/2) * exp(2*n)). - Vaclav Kotesovec, Nov 12 2020

Example

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 + ...
where
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.

Prog

(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] }
for(n=0, 20, print1(a(n), ", "))

Crossrefs

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

Sequence in context: A176585 A086534 A198287 * A268705 A078367 A090306

Adjacent sequences: A338632 A338633 A338634 * A338636 A338637 A338638

Keyword

nonn

Author

Paul D. Hanna, Nov 04 2020

Status

approved