A266526 a(n) = coefficient of x^n*y^(n+1)/n! in Log( Sum_{n>=0} (n + y)^(2*n) * x^n/n! ), for n>=1.

1, 4, 42, 752, 19360, 654912, 27546736, 1388207872, 81621893376, 5488951731200, 415721105434624, 35026876903256064, 3250356630453317632, 329437813126362185728, 36214170617862339840000, 4291812357982293898231808, 545518054282041342531076096, 74032137722410904128877494272, 10684317262536125210489796296704, 1634019721630446295055397683200000

Offset: 1

Author

Paul D. Hanna, Jan 01 2016

Keywords

• nonn

Comments

Equals the logarithm of the e.g.f. of A266481.

Equals the right border of triangle A266521.

Examples

			E.g.f: A(x) = x + 4*x^2/2! + 42*x^3/3! + 752*x^4/4! + 19360*x^5/5! + 654912*x^6/6! + 27546736*x^7/7! + 1388207872*x^8/8! + 81621893376*x^9/9! + 5488951731200*x^10/10! +...
where exponentiation yields the e.g.f. of A266481:
exp(A(x)) = 1 + x + 5*x^2/2! + 55*x^3/3! + 993*x^4/4! + 25501*x^5/5! + 857773*x^6/6! + 35850795*x^7/7! + 1795564865*x^8/8! + 104972371417*x^9/9! +...+ A266481(n)*x^n/n! +...
which equals
Limit_{N->oo} [ Sum_{n>=0} (N + n)^(2*n) * (x/N)^n/n! ]^(1/N).
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..230

Crossrefs

Cf. A266521, A266481.

Programs

PARI

{a(n) = n! * polcoeff( polcoeff( log( sum(m=0,n+1, (m + y)^(2*m) *x^m/m! ) +x*O(x^n) ),n,x), n+1,y)}
for(n=1,30, print1(a(n),", "))

Formula

a(n) ~ 2^(n - 1/4) * (1 + sqrt(2))^(n - 1/2) * exp((1 - sqrt(2))*n) * n^(n-2). - Vaclav Kotesovec, Mar 20 2024