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%I #40 Jan 03 2024 23:23:29
%S 1,2,5,9,18,32,57,95,162,261,418,659,1016,1555,2347,3499,5152,7558,
%T 10914,15704,22363,31684,44460,62161,86191,119026,163282,223015,
%U 302854,409809,551477,739370,987091,1312752,1739064,2295880,3020066,3959580,5175156,6742034
%N Number of Frobenius partitions of 2*n that satisfy the condition that the sum of the entries on the top row plus the number of columns is less than or equal to the sum of the entries on the bottom row.
%H Andrew Howroyd, <a href="/A342208/b342208.txt">Table of n, a(n) for n = 1..1000</a>
%H Kelsey Blum, <a href="https://arxiv.org/abs/2103.03196">Bounds on the Number of Graphical Partitions</a>, arXiv:2103.03196 [math.CO], 2021. See Table on p. 7. [Given a(10) is incorrect.]
%F A000569(n) <= a(n) <= A058696(n). - _Kelsey A. Blum_, Mar 15 2021
%o (PARI) \\ by partitions
%o a(n)={my(total=0); forpart(q=2*n, my(p=Vecrev(q), m=0, s=0); while(m<#p && p[m+1]>m, m++; s+=p[m]-m); if(s + m <= n, total++) ); total} \\ _Andrew Howroyd_, Jan 03 2024
%o (PARI) \\ faster version using g.f.'s
%o a(n)=sum(m=1, sqrtint(2*n), my(r=2*n-m^2); my(g=1/prod(k=1, m, 1 - x^k + O(x*x^r))); sum(i=0, n-binomial(m+1,2), polcoef(g,i)*polcoef(g,r-i)) ) \\ _Andrew Howroyd_, Jan 03 2024
%Y Cf. A000569, A058696.
%K nonn
%O 1,2
%A _Michel Marcus_, Mar 05 2021
%E Corrected and extended by _Andrew Howroyd_, Jan 03 2024
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