A301585 G.f.: Sum_{n>=0} ((1+x)^(3*n) - 1)^n.

1, 3, 39, 910, 29949, 1271751, 66116065, 4066082856, 288701376912, 23240635243591, 2091554595246705, 208085119389952134, 22676957610808295192, 2686515300821612112411, 343760257348413122290260, 47248346582443326267328400, 6942339982115290619799947901, 1085919469129099832397573088863, 180160797497273341662653292624309, 31598815412054398239059538582525618

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..325

Formula

G.f.: Sum_{n>=0} (1+x)^(3*n^2) /(1 + (1+x)^(3*n))^(n+1).

a(n) ~ c * d^n * n! / sqrt(n), where d = 3*A317855 = 9.4832659615962864414905166077643974751791483225656690248818346226130911776579... and c = 0.3108017465925995208675813879173750641359609... - Vaclav Kotesovec, Aug 09 2018

Example

G.f.: A(x) = 1 + 3*x + 39*x^2 + 910*x^3 + 29949*x^4 + 1271751*x^5 + 66116065*x^6 + 4066082856*x^7 + 288701376912*x^8 + ...
such that
A(x) = 1 + ((1+x)^3-1) + ((1+x)^6-1)^2 + ((1+x)^9-1)^3 + ((1+x)^12-1)^4 + ((1+x)^15-1)^5 + ((1+x)^18-1)^6 + ((1+x)^21-1)^7 + ...
Also,
A(x) = 1/2 + (1+x)^3/(1 + (1+x)^3)^2 + (1+x)^12/(1 + (1+x)^6)^3 + (1+x)^27/(1 + (1+x)^9)^4 + (1+x)^48/(1 + (1+x)^12)^5 + (1+x)^75/(1 + (1+x)^15)^6 + ...

Prog

(PARI) {a(n) = my(A, o=x*O(x^n)); A = sum(m=0, n, ((1+x +o)^(3*m) - 1)^m ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A122400, A301584, A301586.

Sequence in context: A121247 A064732 A092610 * A258923 A326271 A276964

Adjacent sequences: A301582 A301583 A301584 * A301586 A301587 A301588

Keyword

nonn

Author

Paul D. Hanna, Mar 24 2018

Status

approved