A276752 G.f.: exp( Sum_{n>=1} [Sum_{k>=1} k^(2*n) * x^k]^n / n ), a power series in x with integer coefficients.

1, 1, 5, 30, 327, 7085, 307280, 28472653, 5000661017, 1886425568702, 1331753751874235, 2008313162512681569, 5765904212733638946976, 34525801618218187545094977, 406111805399407205212602871837, 9635669704681654899673855841540822, 464496624513770925349468939192278531231, 43718131231809168093455159164707384418710045, 8598321846236415035740539472279473819390935625008

Offset

0,3

Comments

Conjecture: a(n)^(1/n^2) tends to sqrt(2). - Vaclav Kotesovec, Oct 17 2020

Links

Paul D. Hanna, Table of n, a(n) for n = 0..100

Formula

G.f.: exp( Sum_{n>=1} [ Sum_{k=1..2*n} A008292(2*n,k) * x^k / (1-x)^(2*n+1) ]^n / n ), where A008292 are the Eulerian numbers.

Example

G.f.: A(x) = 1 + x + 5*x^2 + 30*x^3 + 327*x^4 + 7085*x^5 + 307280*x^6 + 28472653*x^7 + 5000661017*x^8 + 1886425568702*x^9 + 1331753751874235*x^10 +...
log(A(x)) = x + 9*x^2/2 + 76*x^3/3 + 1157*x^4/4 + 33291*x^5/5 + 1792296*x^6/6 + 196919213*x^7/7 + 39766253741*x^8/8 + 16931726147956*x^9/9 + 13298466280839329*x^10/10 + 22076711237844558263*x^11/11 + 69166686377284889199104*x^12/12 +...+ A276754(n)*x^n/n +...
The logarithm of g.f. A(x) equals the series:
log(A(x)) = Sum_{n>=1} (x + 2^(2*n)*x^2 + 3^(2*n)*x^3 +...+ k^(2*n)*x^k +...)^n/n.
This logarithmic series can be written using the Eulerian numbers like so:
log(A(x)) = (x + x^2)/(1-x)^3 + (x + 11*x^2 + 11*x^3 + x^4)^2/(1-x)^10/2 + (x + 57*x^2 + 302*x^3 + 302*x^4 + 57*x^5 + x^6)^3/(1-x)^21/3 + (x + 247*x^2 + 4293*x^3 + 15619*x^4 + 15619*x^5 + 4293*x^6 + 247*x^7 + x^8)^4/(1-x)^36/4 + (x + 1013*x^2 + 47840*x^3 + 455192*x^4 + 1310354*x^5 + 1310354*x^6 + 455192*x^7 + 47840*x^8 + 1013*x^9 + x^10)^5/(1-x)^55/5 + (x + 4083*x^2 + 478271*x^3 + 10187685*x^4 + 66318474*x^5 + 162512286*x^6 + 162512286*x^7 + 66318474*x^8 + 10187685*x^9 + 478271*x^10 + 4083*x^11 + x^12)^6/(1-x)^78/6 +...+ [ Sum_{k=1..2*n} A008292(2*n,k) * x^k ]^n / (1-x)^(2*n^2+n) /n +...

Prog

(PARI) {a(n) = polcoeff( exp( sum(m=1, n+1, sum(k=1, n+1, k^(2*m) * x^k +x*O(x^n))^m / m ) ), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {A008292(n, k) = sum(j=0, k, (-1)^j * (k-j)^n * binomial(n+1, j))}
{a(n) = my(A=1, Oxn=x*O(x^n)); A = exp( sum(m=1, n+1, sum(k=1, 2*m, A008292(2*m, k)*x^k/(1-x +Oxn)^(2*m+1) )^m / m ) ); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A276754, A156170, A276751.

Sequence in context: A344525 A277464 A258300 * A052172 A058783 A203769

Adjacent sequences: A276749 A276750 A276751 * A276753 A276754 A276755

Keyword

nonn

Author

Paul D. Hanna, Sep 16 2016

Status

approved