A301585 - OEIS

%I #13 Aug 09 2018 09:43:26

%S 1,3,39,910,29949,1271751,66116065,4066082856,288701376912,

%T 23240635243591,2091554595246705,208085119389952134,

%U 22676957610808295192,2686515300821612112411,343760257348413122290260,47248346582443326267328400,6942339982115290619799947901,1085919469129099832397573088863,180160797497273341662653292624309,31598815412054398239059538582525618

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

%H Vaclav Kotesovec, <a href="/A301585/b301585.txt">Table of n, a(n) for n = 0..325</a>

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

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

%e 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 + ...

%e such that

%e 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 + ...

%e Also,

%e 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 + ...

%o (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)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A122400, A301584, A301586.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Mar 24 2018