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A237826
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Number of partitions of n such that 4*(least part) = greatest part.
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10
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0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 9, 12, 16, 20, 26, 31, 38, 47, 55, 67, 78, 92, 106, 126, 145, 167, 190, 219, 247, 288, 320, 366, 410, 466, 520, 591, 654, 739, 820, 924, 1018, 1148, 1263, 1415, 1562, 1740, 1911, 2136, 2342, 2607, 2859, 3169, 3469, 3849, 4208
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OFFSET
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1,7
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LINKS
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FORMULA
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G.f.: Sum_{k>=1} x^(5*k)/Product_{j=k..4*k} (1-x^j). - Seiichi Manyama, May 14 2023
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EXAMPLE
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a(8) = 3 counts these partitions: 431, 4211, 41111.
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MATHEMATICA
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z = 64; q[n_] := q[n] = IntegerPartitions[n];
Table[Count[q[n], p_ /; 3 Min[p] == Max[p]], {n, z}] (* A237825*)
Table[Count[q[n], p_ /; 4 Min[p] == Max[p]], {n, z}] (* A237826 *)
Table[Count[q[n], p_ /; 5 Min[p] == Max[p]], {n, z}] (* A237827 *)
Table[Count[q[n], p_ /; 2 Min[p] + 1 == Max[p]], {n, z}] (* A237828 *)
Table[Count[q[n], p_ /; 2 Min[p] - 1 == Max[p]], {n, z}] (* A237829 *)
Table[Count[IntegerPartitions[n], _?(#[[1]]==4#[[-1]]&)], {n, 60}] (* Harvey P. Dale, Jun 15 2023 *)
kmax = 55;
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PROG
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(PARI) my(N=60, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, N, x^(5*k)/prod(j=k, 4*k, 1-x^j)))) \\ Seiichi Manyama, May 14 2023
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CROSSREFS
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KEYWORD
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nonn,easy,changed
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AUTHOR
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STATUS
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approved
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