A183612 - OEIS

%I #10 Jan 04 2020 12:59:11

%S 4,56,1360,47840,2261824,137976704,10568807680,995590733312,

%T 113361706759168,15371500989986816,2450078519983230976,

%U 453832268624393265152,96714634054173633495040,23502233324497426740641792,6461311058964160135965245440,1995601318347535298840189861888

%N Logarithmic derivative of Sum_{n>=0} (n+1)!^2*x^n.

%C Compare with A027837, which is the logarithmic derivative of Sum_{n>=0} n!^2*x^n, and lists the number of subgroups of index n in free group of rank 3.

%H Andrew Howroyd, <a href="/A183612/b183612.txt">Table of n, a(n) for n = 1..200</a>

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

%F G.f.: (Sum_{k>=1} k*(k+1)!^2*x^k)/(Sum_{k>=0} (k+1)!^2*x^k). - _Andrew Howroyd_, Jan 04 2020

%e G.f.: A(x) = 4*x + 56*x^2/2 + 1360*x^3/3 + 47840*x^4/4 +... where

%e exp(A(x)) = 1 + 4*x + 36*x^2 + 576*x^3 + 14400*x^4 +...+ (n+1)!^2*x^n +...

%o (PARI) {a(n)=if(n<1,0,polcoeff(x*deriv(log(sum(m=0,n,(m+1)!^2*x^m)+x*O(x^n))),n))}

%o (PARI) seq(n) = Vec((sum(k=1, n, k*(k+1)!^2*x^k))/(sum(k=0, n, (k+1)!^2*x^k)) + O(x*x^n)) \\ _Andrew Howroyd_, Jan 04 2020

%Y Cf. A027837.

%K nonn

%O 1,1

%A _Paul D. Hanna_, Jan 16 2011

%E Terms a(14) and beyond from _Andrew Howroyd_, Jan 03 2020