A183612 Logarithmic derivative of Sum_{n>=0} (n+1)!^2*x^n.

4, 56, 1360, 47840, 2261824, 137976704, 10568807680, 995590733312, 113361706759168, 15371500989986816, 2450078519983230976, 453832268624393265152, 96714634054173633495040, 23502233324497426740641792, 6461311058964160135965245440, 1995601318347535298840189861888

Offset

1,1

Comments

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.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Formula

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

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

Example

G.f.: A(x) = 4*x + 56*x^2/2 + 1360*x^3/3 + 47840*x^4/4 +... where
exp(A(x)) = 1 + 4*x + 36*x^2 + 576*x^3 + 14400*x^4 +...+ (n+1)!^2*x^n +...

Prog

(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))}
(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

Crossrefs

Cf. A027837.

Sequence in context: A361541 A218309 A061924 * A277038 A009536 A009558

Adjacent sequences: A183609 A183610 A183611 * A183613 A183614 A183615

Keyword

nonn

Author

Paul D. Hanna, Jan 16 2011

Extensions

Terms a(14) and beyond from Andrew Howroyd, Jan 03 2020

Status

approved