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A192478
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G.f. satisfies: A(x) = x*Product_{n>=1} 1/(1 - x*A(x)^n).
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1
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1, 0, 1, 1, 3, 5, 14, 28, 75, 170, 443, 1076, 2795, 7046, 18398, 47458, 124793, 326905, 865853, 2293103, 6114688, 16327765, 43800590, 117720693, 317452606, 857742087, 2323588888, 6306229600, 17151172495, 46725729232, 127522337636, 348562660876
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OFFSET
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1,5
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COMMENTS
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Related q-series identity (Euler):
Product_{n>=1} 1/(1-x*q^n) = Sum_{n>=0} x^n*q^n / Product_{k=1..n} (1-q^k); here q=A(x).
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LINKS
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Table of n, a(n) for n=1..32.
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FORMULA
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G.f. satisfies: A(x) = x*Sum_{n>=0} x^n*A(x)^n / Product_{k=1..n} (1 - A(x)^k).
G.f. satisfies: A(x) = x*Sum_{n>=0} x^n*A(x)^(n^2) / (Product_{k=1..n} (1 - A(x)^k)*(1 - x*A(x)^k)) due to Cauchy's identity.
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EXAMPLE
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G.f.: A(x) = x + x^3 + x^4 + 3*x^5 + 5*x^6 + 14*x^7 + 28*x^8 + 75*x^9 +...
The g.f. A = A(x) satisfies the relations:
A = x/((1 - x*A)*(1 - x*A^2)*(1 - x*A^3)*(1 - x*A^4)*...),
A = x*(1 + x*A/(1-A) + x^2*A^2/((1-A)*(1-A^2)) + x^3*A^3/((1-A)*(1-A^2)*(1-A^3)) +...).
A = x*(1 + x*A/((1-A)*(1-x*A)) + x^2*A^4/((1-A)*(1-x*A)*(1-A^2)*(1-x*A^2)) + x^3*A^9/((1-A)*(1-x*A)*(1-A^2)*(1-x*A^2)*(1-A^3)*(1-x*A^3)) +...).
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PROG
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(PARI) {a(n)=local(A=x+x^2); for(i=1, n, A=x/prod(m=1, n, (1-x*A^m +x*O(x^n) ))); polcoeff(A, n)}
(PARI) {a(n)=local(A=x+x^2); for(i=1, n, A=x*sum(m=0, n, x^m*A^m/prod(k=1, m, (1-A^k +x*O(x^n) )))); polcoeff(A, n)}
(PARI) {a(n)=local(A=x+x^2); for(i=1, n, A=x*sum(m=0, sqrtint(n+1), A^(m^2)*x^m/prod(k=1, m, (1-A^k)*(1-x^k*A +x*O(x^n) )))); polcoeff(A, n)}
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CROSSREFS
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Cf. A145268, A192477.
Sequence in context: A007136 A145974 A147544 * A198785 A222380 A271867
Adjacent sequences: A192475 A192476 A192477 * A192479 A192480 A192481
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna, Jul 01 2011
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STATUS
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approved
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