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.

1, 2, 5, 9, 18, 32, 57, 95, 162, 261, 418, 659, 1016, 1555, 2347, 3499, 5152, 7558, 10914, 15704, 22363, 31684, 44460, 62161, 86191, 119026, 163282, 223015, 302854, 409809, 551477, 739370, 987091, 1312752, 1739064, 2295880, 3020066, 3959580, 5175156, 6742034

Offset: 1

Author

Michel Marcus, Mar 05 2021

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000
• Kelsey Blum, Bounds on the Number of Graphical Partitions, arXiv:2103.03196 [math.CO], 2021. See Table on p. 7. [Given a(10) is incorrect.]

Crossrefs

Cf. A000569, A058696.

Programs

PARI

\\ by partitions
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

PARI

\\ faster version using g.f.'s
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

Formula

A000569(n) <= a(n) <= A058696(n). - Kelsey A. Blum, Mar 15 2021

Extensions

Corrected and extended by Andrew Howroyd, Jan 03 2024