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A195983
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G.f.: Product_{n>=1} [Product_{k=0..2*n} (1 - k*x)^(-(-1)^k*binomial(2*n,k))]^(1/n).
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2
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1, 0, 1, 2, 7, 32, 189, 1382, 12090, 123220, 1433760, 18750856, 272243150, 4344694984, 75590125814, 1423923579636, 28872306351182, 626986388773656, 14518030696953650, 357068944503961140, 9296168815245928682, 255409012110904594768, 7385107814017356745382
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OFFSET
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0,4
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COMMENTS
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Compare g.f. to the product:
_ exp(x) = Product_{n>=1} [Product_{k=0..n} (1 + k*x)^(-(-1)^k*binomial(n,k))]^(1/n),
more explicitly:
_ exp(x) = [(1+x)^1/(1+0*x)^1] * [(1+x)^2/((1+0*x)^1*(1+2*x)^1)]^(1/2) * [(1+x)^3*(1+3*x)^1/((1+0*x)^1*(1+2*x)^3)]^(1/3) * [(1+x)^4*(1+3*x)^4/((1+0*x)^1*(1+2*x)^6*(1+4*x)^1)]^(1/4) * ...
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LINKS
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FORMULA
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G.f.: A(x) = exp( Sum_{n>=1} 2*A000670(n)*x^(n+1)/(n+1) ) where Sum_{n>=0} A000670(n)*x^n = Sum_{n>=0} n!*x^n/Product_{k=0..n} (1-k*x).
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EXAMPLE
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G.f.: A(x) = 1 + x^2 + 2*x^3 + 7*x^4 + 32*x^5 + 189*x^6 + 1382*x^7 +...
where
A(x) = [(1-x)^2/((1-0*x)^1*(1-2*x)^1)] * [(1-x)^4*(1-3*x)^4/((1-0*x)^1*(1-2*x)^6*(1-4*x)^1)]^(1/2) * [(1-x)^6*(1-3*x)^20*(1-5*x)^6/((1-0*x)^1*(1-2*x)^15*(1-4*x)^15*(1-6*x)^1)]^(1/3) * ...
Also,
log(A(x)) = 2*x^2/2 + 6*x^3/3 + 26*x^4/4 + 150*x^5/5 + 1082*x^6/6 + 9366*x^7/7 +...+ 2*A000670(n)*x^(n+1)/(n+1) +...
thus, the logarithmic derivative equals the series:
A'(x)/A(x) = 2*x/(1-x) + 2*2!*x^2/((1-x)*(1-2*x)) + 2*3!*x^3/((1-x)*(1-2*x)*(1-3*x)) + 2*4!*x^4/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)) +...
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PROG
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(PARI) {a(n)=polcoeff(prod(m=1, n, prod(k=1, 2*m, (1-k*x+x*O(x^n))^(-(-1)^k*binomial(2*m, k)/m))), n)}
(PARI) /* Using o.g.f. exp( Sum_{n>=1} 2*A000670(n)*x^(n+1)/(n+1) ): */
{a(n)=polcoeff(exp(intformal(2*sum(m=1, n+1, m!*x^m/prod(k=1, m, 1-k*x+x*O(x^n))))), 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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