A277037 - OEIS

%I #19 Oct 08 2016 19:47:34

%S 1,2,18,484,54630,26638924,53843811956,442942117297000,

%T 14725418961500037126,1971239927985067569365772,

%U 1060292226589575099894174194524,2288290973515256950275126683431946552,19795837218795604674370624304477542380054748,685985356865646724678258830150265065104998427771576,95174256167264272421248219248338459257647770713814222870312

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

%C More generally, for fixed integer q, G(x,q) = exp( Sum_{n>=1} [Sum_{k>=1} k^n * q^(n*k) * x^k]^n / n ) is an integer series such that G(x,q) = exp( Sum_{n>=1} q^(n^2)/(1 - q^n*x)^(n^2+n) * [ Sum_{k=1..n} A008292(n,k) * q^(n*k-n) * x^k ]^n / n ), where A008292 are the Eulerian numbers.

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

%e G.f.: A(x) = 1 + 2*x + 18*x^2 + 484*x^3 + 54630*x^4 + 26638924*x^5 + 53843811956*x^6 + 442942117297000*x^7 +...

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

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

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

%e log(A(x)) = 2*x/(1-2*x)^2 + 2^4*(x + 2^2*x^2)^2/(1-2^2*x)^6/2 + 2^9*(x + 4*2^3*x^2 + 2^6*x^3)^3/(1-2^3*x)^12/3 + 2^16*(x + 11*2^4*x^2 + 11*2^8*x^3 + 2^24*x^4)^4/(1-2^4*x)^20/4 + 2^25*(x + 26*2^5*x^2 + 66*2^10*x^3 + 26*2^15*x^4 + 2^20*x^5)^5/(1-2^5*x)^30/5 + 2^36*(x + 57*2^6*x^2 + 302*2^12*x^3 + 302*2^18*x^4 + 57*2^24*x^5 + 2^30*x^6)^6/(1-2^6*x)^42/6 +...+ [ Sum_{k=1..n} A008292(n,k) * 2^(n*k) * x^k ]^n / (1 - 2^n*x)^(n*(n+1))/n +...

%e Explicitly,

%e log(A(x)) = 2*x + 32*x^2/2 + 1352*x^3/3 + 214272*x^4/4 + 132616992*x^5/5 + 322738100480*x^6/6 + 3099838240135296*x^7/7 + 117796258487089512448*x^8/8 +...

%o (PARI) {a(n) = my(A=1,Oxn=x*O(x^n)); A = exp( sum(m=1,n+1, sum(k=1,n+1, k^m * 2^(m*k) * x^k +x*O(x^n) )^m / m )); polcoeff(A,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, m, A008292(m, k) * 2^(m*k) * x^k / (1 - 2^m*x +Oxn)^(m+1) )^m / m ) ); polcoeff(A, n)}

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

%Y Cf. A277035, A156170, A277036, A008292.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 08 2016