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A277037
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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.
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2
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1, 2, 18, 484, 54630, 26638924, 53843811956, 442942117297000, 14725418961500037126, 1971239927985067569365772, 1060292226589575099894174194524, 2288290973515256950275126683431946552, 19795837218795604674370624304477542380054748, 685985356865646724678258830150265065104998427771576, 95174256167264272421248219248338459257647770713814222870312
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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 +...
such that the logarithm of g.f. A(x) equals the series:
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.
This logarithmic series can be written using the Eulerian numbers like so:
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 +...
Explicitly,
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 +...
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PROG
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(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)}
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, m, A008292(m, k) * 2^(m*k) * x^k / (1 - 2^m*x +Oxn)^(m+1) )^m / m ) ); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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