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A237829
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Number of partitions of n such that 2*(least part) - 1 = greatest part.
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5
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1, 1, 1, 1, 2, 1, 2, 3, 2, 2, 5, 3, 4, 5, 5, 6, 8, 6, 8, 10, 10, 10, 15, 12, 14, 17, 18, 20, 23, 21, 26, 29, 30, 31, 39, 38, 42, 46, 49, 52, 61, 60, 68, 74, 77, 83, 94, 95, 104, 112, 122, 128, 143, 144, 159, 172, 181, 192, 212, 219, 237, 253, 271, 285
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OFFSET
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1,5
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LINKS
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FORMULA
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G.f.: x + Sum_{k>=1} x^(3*k-1)/Product_{j=k..2*k-1} (1-x^j). - Seiichi Manyama, May 17 2023
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EXAMPLE
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a(8) = 3 counts these partitions: 53, 332, 11111111.
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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 *)
(* Second program: *)
kmax = 64;
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PROG
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(PARI) my(N=70, x='x+O('x^N)); Vec(x+sum(k=1, N, x^(3*k-1)/prod(j=k, 2*k-1, 1-x^j))) \\ Seiichi Manyama, May 17 2023
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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