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A168657
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Number of partitions of n such that the number of parts is divisible by the smallest part.
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9
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1, 1, 2, 4, 6, 8, 12, 17, 25, 34, 48, 64, 87, 114, 151, 198, 258, 332, 428, 546, 695, 879, 1108, 1388, 1737, 2159, 2680, 3312, 4082, 5009, 6138, 7492, 9126, 11081, 13429, 16228, 19575, 23547, 28277, 33879, 40520, 48354, 57615, 68509, 81337, 96388, 114055
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OFFSET
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1,3
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LINKS
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FORMULA
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G.f.: Sum_{n>=1} Sum_{d|n} x^(n*d)/Product_{k=1..n-1}(1-x^k).
G.f.: Sum_{i>=1} Sum_{j>=1} x^(i*j^2)/Product_{k=1..i*j-1} (1-x^k). - Seiichi Manyama, Jan 21 2022
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MAPLE
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b:= proc(n, i, t) option remember;
`if`(n<1, 0, `if`(i=1, 1, `if`(i<1, 0,
`if`(irem(n, i)=0 and irem(t+n/i, i)=0, 1, 0)+
add(b(n-i*j, i-1, t+j), j=0..n/i))))
end:
a:= n-> b(n, n, 0):
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MATHEMATICA
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b[n_, i_, t_] := b[n, i, t] = If[n<1, 0, If[i==1, 1, If[i<1, 0, If [Mod[n, i]==0 && Mod[t+n/i, i]==0, 1, 0] + Sum[b[n-i*j, i-1, t+j], {j, 0, n/i}]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Jul 01 2015, after Alois P. Heinz *)
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PROG
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(PARI) my(N=66, x='x+O('x^N)); Vec(sum(i=1, N, sum(j=1, sqrtint(N\i), x^(i*j^2)/prod(k=1, i*j-1, 1-x^k)))) \\ Seiichi Manyama, Jan 21 2022
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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