|
%I #43 Jan 05 2022 13:53:30
%S 1,2,6,16,38,88,196,420,878,1794,3584,7032,13572,25792,48352,89512,
%T 163774,296444,531234,943072,1659560,2896376,5015700,8622108,14718652,
%U 24960138,42062200,70458160,117349856,194381704,320295312,525123604
%N G.f.: Product_{n>0} ((1+x^n)/(1-x^n))^n.
%C Generating function for a sum over strict plane partitions weighted with 2 powered to their number of connected components.
%C 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
%C 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
%C 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
%H Seiichi Manyama, <a href="/A156616/b156616.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Vaclav Kotesovec)
%H Ali H. Al-Saedi, <a href="https://doi.org/10.1007/s11139-018-0036-5">Congruences for restricted plane overpartitions modulo 4 and 8</a>, Raman. J. 48 (2) (2019) 251
%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 18.
%H Mirjana Vuletic, <a href="http://dx.doi.org/10.1090/S0002-9947-08-04753-3">A generalization of MacMahon's formula</a>, Trans. Am. Math. Soc. 361 (2009) 2789-2804.
%F Convolve A000219 with A026007.
%F 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
%F 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
%F a(0) = 1, a(n) = (2/n)*Sum_{k=1..n} A076577(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Apr 30 2017
%F 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
%t nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 17 2015 *)
%o (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
%Y Cf. A000219, A015128, A026007, A261384, A261386, A261389.
%Y Cf. A206622, A206623, A206624, A261519, A261520.
%Y Cf. A285462, A285447, A285460, A285461.
%Y Cf. A285446, A285458, A285459.
%Y Cf. A306081.
%K easy,nonn
%O 0,2
%A _R. J. Mathar_, Feb 11 2009
|
