A276752 - OEIS

%I #17 Oct 17 2020 07:40:04

%S 1,1,5,30,327,7085,307280,28472653,5000661017,1886425568702,

%T 1331753751874235,2008313162512681569,5765904212733638946976,

%U 34525801618218187545094977,406111805399407205212602871837,9635669704681654899673855841540822,464496624513770925349468939192278531231,43718131231809168093455159164707384418710045,8598321846236415035740539472279473819390935625008

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

%C Conjecture: a(n)^(1/n^2) tends to sqrt(2). - _Vaclav Kotesovec_, Oct 17 2020

%H Paul D. Hanna, <a href="/A276752/b276752.txt">Table of n, a(n) for n = 0..100</a>

%F 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.

%e 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 +...

%e 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 +...

%e The logarithm of g.f. A(x) equals the series:

%e log(A(x)) = Sum_{n>=1} (x + 2^(2*n)*x^2 + 3^(2*n)*x^3 +...+ k^(2*n)*x^k +...)^n/n.

%e This logarithmic series can be written using the Eulerian numbers like so:

%e 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 +...

%o (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)}

%o for(n=0, 20, print1(a(n), ", "))

%o (PARI) {A008292(n, k) = sum(j=0, k, (-1)^j * (k-j)^n * binomial(n+1, j))}

%o {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)}

%o for(n=0, 20, print1(a(n), ", "))

%Y Cf. A276754, A156170, A276751.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 16 2016