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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.
1
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
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
Sequence in context: A288578 A002883 A295724 * A077865 A117353 A103422
KEYWORD
nonn
AUTHOR
Michel Marcus, Mar 05 2021
EXTENSIONS
Corrected and extended by Andrew Howroyd, Jan 03 2024
STATUS
approved