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A156616
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G.f.: Product_{n>0} ((1+x^n)/(1-x^n))^n.
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51
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1, 2, 6, 16, 38, 88, 196, 420, 878, 1794, 3584, 7032, 13572, 25792, 48352, 89512, 163774, 296444, 531234, 943072, 1659560, 2896376, 5015700, 8622108, 14718652, 24960138, 42062200, 70458160, 117349856, 194381704, 320295312, 525123604
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OFFSET
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0,2
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COMMENTS
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Generating function for a sum over strict plane partitions weighted with 2 powered to their number of connected components.
The inverse Euler transform is apparently 2, 3, 6, 6, 10, 9, 14, 12, 18, 15, 22, 18, 26, 21, ..., A016825 interlaced with A008585. - R. J. Mathar, Apr 23 2009
In general, for m >= 1, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^(m*k), then a(n) ~ exp(m/12 + 3/2 * (7*m*Zeta(3)/2)^(1/3) * n^(2/3)) * m^(1/6 + m/36) * (7*Zeta(3))^(1/6 + m/36) / (A^m * 2^(2/3 + m/9) * sqrt(3*Pi) * n^(2/3 + m/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 17 2015
In general, for m >= 0, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^(k^m), then a(n) ~ ((2^(m+2)-1) * Gamma(m+2) * Zeta(m+2) / (2^(2*m+3) * n))^((1-2*Zeta(-m))/(2*m+4)) * exp((m+2)/(m+1) * ((2^(m+2)-1) * n^(m+1) * Gamma(m+2) * Zeta(m+2) / 2^(m+1))^(1/(m+2)) + Zeta'(-m)) / sqrt((m+2)*Pi*n). - Vaclav Kotesovec, Aug 19 2015
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LINKS
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FORMULA
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O.g.f.: exp( Sum_{n>=1} (sigma_2(2n) - sigma_2(n))/2 *x^n/n ), where sigma_2(n) is the sum of squares of divisors of n (A001157). - Paul D. Hanna, May 01 2010
a(n) ~ exp(1/12 + 3 * 2^(-4/3) * (7*Zeta(3))^(1/3) * n^(2/3)) * (7*Zeta(3))^(7/36) / (A * 2^(7/9) * sqrt(3*Pi) * n^(25/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 17 2015
G.f.: A(x) = exp( 2*Sum_{n >= 0} x^(2*n+1)/((2*n+1)*(1 - x^(2*n+1))^2) ). Cf. A000122 and A302237. - Peter Bala, Dec 23 2021
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MATHEMATICA
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nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 17 2015 *)
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PROG
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(PARI) {a(n)=polcoeff(exp(sum(m=1, n, (sigma(2*m, 2)-sigma(m, 2))/2*x^m/m)+x*O(x^n)), n)} \\ Paul D. Hanna, May 01 2010
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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