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%I #32 Jun 07 2018 12:54:09
%S 1,2,14,154,2062,31066,504886,8652402,154208270,2832526306,
%T 53287424374,1022143389578,19924535352374,393685747760714,
%U 7869272950148382,158875743754158098,3235672769357219854,66405081412501161442,1372115409786911859502,28524372351269271839610
%N Number of overpartitions of n^2; a(n) = A015128(n^2).
%C Limit a(n+1)/a(n) = exp(Pi) = 23.14069263...
%C a(n) ~ (cosh(Pi*n) - sinh(Pi*n)/(Pi*n)) / (4*n^2), a "remarkable approximation" due to "Ramanujan's false statement" (see formula 12 in "Jagged partitions" link).
%C By definition of A015128, an overpartition of n^2 is an ordered sequence of nonincreasing integers that sum to n^2, where the first occurrence of each integer may be overlined (see Hirschhorn and Sellers link).
%H Vaclav Kotesovec, <a href="/A219430/b219430.txt">Table of n, a(n) for n = 0..730</a> (terms 0..180 from Paul D. Hanna)
%H J.-F. Fortin, P. Jacob and P. Mathieu, <a href="http://arXiv.org/abs/math.CO/0310079">Jagged partitions</a>
%H M. D. Hirschhorn and J. A. Sellers, <a href="http://www.emis.de/journals/INTEGERS/papers/f20/f20.pdf">AN INFINITE FAMILY OF OVERPARTITION CONGRUENCES MODULO 12</a>
%F a(n) = -2*Sum_{k=1..n} (-1)^k * A015128(n^2-k^2) for n>0 with a(0)=1.
%F a(n) = [x^(n^2)] 1 / ( Sum_{m=-inf..inf} (-x)^(m^2) ).
%F a(n) = [x^(n^2)] 1 / theta_4(x).
%F a(n) = [x^(n^2)] eta(x^2) / eta(x)^2.
%F a(n) = [x^(n^2)] Product_{m>=1} (1 + x^m) / (1 - x^m).
%F a(n) = [x^(n^2)] Product_{m>=1} 1 / ( (1 - x^(2*m)) * (1 - x^(2*m-1))^2 ).
%F a(n) = [x^(n^2)] exp( Sum_{m>=1} 2*x^(2*m-1)/(1 - x^(2*m-1))/(2*m-1) ).
%F a(n) = [x^(n^2)] exp( Sum_{m>=1} (sigma(2*m) - sigma(m)) * x^m/m ).
%e G.f.: A(x) = 1 + 2*x + 14*x^2 + 154*x^3 + 2062*x^4 + 31066*x^5 + 504886*x^6 +...
%e It appears that the logarithmic derivative of the g.f. A(x),
%e A'(x)/A(x) = 2 + 24*x + 386*x^2 + 6832*x^3 + 128442*x^4 + 2505720*x^5 + 50153770*x^6 + 1022997344*x^7 + 21170657906*x^8 +...+ A219431(n+1)*x^n +...
%e is congruent to 2/(1-x^2) mod 4.
%t Table[Sum[PartitionsP[n^2-k]*PartitionsQ[k], {k, 0, n^2}], {n, 0, 20}] (* _Vaclav Kotesovec_, Nov 28 2015 *)
%o (PARI) /* Formula: a(n) = [x^(n^2)] 1 / theta_4(x) */
%o {a(n)=polcoeff(1/(1+2*sum(k=1,n,(-x)^(k^2))+x*O(x^(n^2))),n^2)}
%o for(n=0,20,print1(a(n),", "))
%o (PARI) /* Formula: a(n) = -2*Sum_{k=1..n} (-1)^k * A015128(n^2-k^2) */
%o {A015128(n)=polcoeff(1/(1+2*sum(k=1, sqrtint(n+1), (-x)^(k^2))+x*O(x^(n))), n)}
%o {a(n)=if(n==0,1,-2*sum(k=1, n, (-1)^k*A015128(n^2-k^2)))}
%o for(n=0, 25, print1(a(n), ", "))
%Y Cf. A219431, A015128, A195584.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Nov 19 2012
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