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A238864
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Number of partitions of n where the difference between consecutive parts is at most 4.
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10
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1, 1, 2, 3, 5, 7, 11, 14, 20, 26, 36, 46, 63, 79, 104, 131, 169, 210, 269, 332, 418, 515, 640, 782, 967, 1173, 1435, 1736, 2108, 2534, 3062, 3663, 4398, 5243, 6259, 7429, 8834, 10441, 12356, 14559, 17159, 20144, 23661, 27686, 32403, 37807, 44102, 51306, 59680, 69235, 80297, 92924, 107482, 124070, 143157, 164862, 189763, 218057
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OFFSET
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0,3
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COMMENTS
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Also the number of partitions of n such that all parts, with the possible exception of the largest are repeated at most four times (by taking conjugates).
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LINKS
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FORMULA
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G.f.: 1 + sum(k>=1, q^k/(1-q^k) * prod(i=1..k-1, (1-q^(5*i))/(1-q^i) ) ).
a(n) ~ exp(Pi*sqrt(8*n/15)) / (3^(1/4) * 10^(3/4) * n^(3/4)). - Vaclav Kotesovec, Jan 26 2022
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(b(n-i*j, i-1), j=0..min(4, n/i))))
end:
g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(b(n-i*j, i-1), j=1..n/i)))
end:
a:= n-> add(g(n, k), k=0..n):
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MATHEMATICA
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b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i-1], {j, 0, Min[4, n/i]}]]]; g[n_, i_] := g[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n - i*j, i - 1], {j, 1, n/i}]]]; a[n_] := Sum[g[n, k], {k, 0, n}]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *)
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PROG
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(PARI)
N=66; q = 'q + O('q^N);
Vec( 1 + sum(k=1, N, q^k/(1-q^k) * prod(i=1, k-1, (1-q^(5*i))/(1-q^i) ) ) )
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CROSSREFS
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Sequences "number of partitions with max diff d": A000005 (d=0, for n>=1), A034296 (d=1), A224956 (d=2), A238863 (d=3), this sequence, A238865 (d=5), A238866 (d=6), A238867 (d=7), A238868 (d=8), A238869 (d=9), A000041 (d --> infinity).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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