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

1,2

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.]

Formula

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

Prog

(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

Crossrefs

Cf. A000569, A058696.

Sequence in context: A288578 A002883 A295724 * A077865 A117353 A103422

Adjacent sequences: A342205 A342206 A342207 * A342209 A342210 A342211

Keyword

nonn

Author

Michel Marcus, Mar 05 2021

Extensions

Corrected and extended by Andrew Howroyd, Jan 03 2024

Status

approved