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%I #7 Oct 08 2016 19:35:27
%S 1,2,14,320,21036,4248736,2753284608,5889659239296,42571528094271584,
%T 1060699597956427433984,92622122614950875482410496,
%U 28732153964162783015337150191616,32013867511269166370870196132112760832,129287051721999031624124705228031781712207872,1906782843976072893849368486957954962408685271556096
%N G.f.: Sum_{n>=0} log(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) = Sum_{n>=0} log(1 + Sum_{k>=1} k^n * q^(n*k) * x^k )^n / n! is an integer series such that G(x,q) = Sum_{n>=0} log(1 + [Sum_{k=1..n} A008292(n,k) * q^(n*k) * x^k]/(1 - q^n*x)^(n+1) )^n / n!, where A008292 are the Eulerian numbers.
%F G.f.: Sum_{n>=0} log(1 + [Sum_{k=1..n} A008292(n,k) * 2^(n*k) * x^k]/(1 - 2^n*x)^(n+1) )^n / n!, where A008292 are the Eulerian numbers.
%e G.f.: A(x) = 1 + 2*x + 14*x^2 + 320*x^3 + 21036*x^4 + 4248736*x^5 + 2753284608*x^6 + 5889659239296*x^7 + 42571528094271584*x^8 +...
%e such that
%e A(x) = Sum_{n>=0} log(1 + 1^n*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 Equivalently,
%e A(x) = 1 + log(1 + 1*2*x + 2*2^2*x^2 + 3*2^3*x^3 + 4*2^4*x^4 +...) +
%e log(1 + 1^2*2^2*x + 2^2*2^4*x^2 + 3^2*2^6*x^3 + 4^2*2^8*x^4 +...)^2/2! +
%e log(1 + 1^3*2^3*x + 2^3*2^6*x^2 + 3^3*2^9*x^3 + 4^3*2^12*x^4 +...)^3/3! +
%e log(1 + 1^4*2^4*x + 2^4*2^8*x^2 + 3^4*2^12*x^3 + 4^4*2^16*x^4 +...)^4/4! +
%e ...
%e The g.f. can be written using the Eulerian numbers like so:
%e A(x) = 1 + log(1 + 2*x/(1-2*x)^2) + log(1 + 2^2*(x + 2^2*x^2)/(1-2^2*x)^3)^2/2! + log(1 + 2^3*(x + 4*2^3*x^2 + 2^6*x^3)/(1-2^3*x)^4)^3/3! + log(1 + 2^4*(x + 11*2^4*x^2 + 11*2^8*x^3 + 2^24*x^4)/(1-2^4*x)^5)^4/4! + log(1 + 2^5*(x + 26*2^5*x^2 + 66*2^10*x^3 + 26*2^15*x^4 + 2^20*x^5)/(1-2^5*x)^6)^5/5! + log(1 + 2^6*(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)/(1-2^6*x)^7)^6/6! +...+ log(1 + Sum_{k=1..n} A008292(n,k) * 2^(n*k) * x^k / (1 - 2^n*x)^(n+1) )^n/n! +...
%o (PARI) {a(n) = my(A=1, Oxn=x*O(x^n));
%o A = sum(m=0, n+1, log(1 + sum(k=1, n+1, k^m * 2^(m*k) * x^k +x*O(x^n)) )^m / m! );
%o 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));
%o A = sum(m=0, n+1, log(1 + sum(k=1, m+1, A008292(m,k) * 2^(m*k) * x^k) / (1 - 2^m*x +Oxn)^(m+1) )^m / m! );
%o polcoeff(A, n)}
%o for(n=0, 20, print1(a(n), ", "))
%Y f. A277037, A277036, A008292.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Oct 08 2016
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