A219430 Number of overpartitions of n^2; a(n) = A015128(n^2).

1, 2, 14, 154, 2062, 31066, 504886, 8652402, 154208270, 2832526306, 53287424374, 1022143389578, 19924535352374, 393685747760714, 7869272950148382, 158875743754158098, 3235672769357219854, 66405081412501161442, 1372115409786911859502, 28524372351269271839610

Offset: 0

Author

Paul D. Hanna, Nov 19 2012

Keywords

• nonn

Comments

Limit a(n+1)/a(n) = exp(Pi) = 23.14069263...

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).

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).

Examples

			G.f.: A(x) = 1 + 2*x + 14*x^2 + 154*x^3 + 2062*x^4 + 31066*x^5 + 504886*x^6 +...
It appears that the logarithmic derivative of the g.f. A(x),
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 +...
is congruent to 2/(1-x^2) mod 4.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..730 (terms 0..180 from Paul D. Hanna)
• J.-F. Fortin, P. Jacob and P. Mathieu, Jagged partitions
• M. D. Hirschhorn and J. A. Sellers, AN INFINITE FAMILY OF OVERPARTITION CONGRUENCES MODULO 12

Crossrefs

Cf. A219431, A015128, A195584.

Programs

Mathematica

Table[Sum[PartitionsP[n^2-k]*PartitionsQ[k], {k, 0, n^2}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 28 2015 *)

PARI

/* Formula: a(n) = [x^(n^2)] 1 / theta_4(x) */
{a(n)=polcoeff(1/(1+2*sum(k=1,n,(-x)^(k^2))+x*O(x^(n^2))),n^2)}
for(n=0,20,print1(a(n),", "))

PARI

/* Formula: a(n) = -2*Sum_{k=1..n} (-1)^k * A015128(n^2-k^2) */
{A015128(n)=polcoeff(1/(1+2*sum(k=1, sqrtint(n+1), (-x)^(k^2))+x*O(x^(n))), n)}
{a(n)=if(n==0,1,-2*sum(k=1, n, (-1)^k*A015128(n^2-k^2)))}
for(n=0, 25, print1(a(n), ", "))

Formula

a(n) = -2*Sum_{k=1..n} (-1)^k * A015128(n^2-k^2) for n>0 with a(0)=1.

a(n) = [x^(n^2)] 1 / ( Sum_{m=-inf..inf} (-x)^(m^2) ).

a(n) = [x^(n^2)] 1 / theta_4(x).

a(n) = [x^(n^2)] eta(x^2) / eta(x)^2.

a(n) = [x^(n^2)] Product_{m>=1} (1 + x^m) / (1 - x^m).

a(n) = [x^(n^2)] Product_{m>=1} 1 / ( (1 - x^(2*m)) * (1 - x^(2*m-1))^2 ).

a(n) = [x^(n^2)] exp( Sum_{m>=1} 2*x^(2*m-1)/(1 - x^(2*m-1))/(2*m-1) ).

a(n) = [x^(n^2)] exp( Sum_{m>=1} (sigma(2*m) - sigma(m)) * x^m/m ).