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A237826
Number of partitions of n such that 4*(least part) = greatest part.
10
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
OFFSET
1,7
LINKS
FORMULA
G.f.: Sum_{k>=1} x^(5*k)/Product_{j=k..4*k} (1-x^j). - Seiichi Manyama, May 14 2023
EXAMPLE
a(8) = 3 counts these partitions: 431, 4211, 41111.
MATHEMATICA
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;
Sum[x^(5k)/Product[1 - x^j, {j, k, 4 k}], {k, 1, kmax}]/x + O[x]^kmax // CoefficientList[#, x]& (* Jean-François Alcover, May 30 2024, after Seiichi Manyama *)
PROG
(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
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 16 2014
STATUS
approved