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A342208 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.

%I A342208 #40 Jan 03 2024 23:23:29
%S A342208 1,2,5,9,18,32,57,95,162,261,418,659,1016,1555,2347,3499,5152,7558,
%T A342208 10914,15704,22363,31684,44460,62161,86191,119026,163282,223015,
%U A342208 302854,409809,551477,739370,987091,1312752,1739064,2295880,3020066,3959580,5175156,6742034
%N A342208 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 A342208 Andrew Howroyd, <a href="/A342208/b342208.txt">Table of n, a(n) for n = 1..1000</a>
%H A342208 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 A342208 A000569(n) <= a(n) <= A058696(n). - _Kelsey A. Blum_, Mar 15 2021
%o A342208 (PARI) \\ by partitions
%o A342208 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 A342208 (PARI) \\ faster version using g.f.'s
%o A342208 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 A342208 Cf. A000569, A058696.
%K A342208 nonn
%O A342208 1,2
%A A342208 _Michel Marcus_, Mar 05 2021
%E A342208 Corrected and extended by _Andrew Howroyd_, Jan 03 2024