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A342208
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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.
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1
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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
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OFFSET
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1,2
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LINKS
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FORMULA
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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