A237828 Number of partitions of n such that 2*(least part) + 1 = greatest part.

0, 0, 0, 1, 1, 2, 4, 4, 6, 9, 10, 12, 17, 18, 22, 27, 31, 34, 42, 45, 53, 61, 66, 72, 86, 92, 103, 113, 125, 135, 154, 163, 180, 197, 213, 229, 257, 271, 294, 318, 346, 368, 404, 426, 463, 497, 532, 564, 616, 651, 700, 747, 798, 844, 912, 962, 1033, 1097, 1167, 1231, 1327, 1397, 1486, 1576, 1677

Offset

1,6

Comments

Also, the number of partitions p of n such that if h = max(p), then h is an (h,0)-separator of p; for example, a(10) counts these 9 partitions: 181, 271, 361, 262, 451, 352, 343, 23131, 1212121. - Clark Kimberling, Mar 24 2014

Links

Table of n, a(n) for n=1..65.

Formula

G.f.: Sum_{k>=1} x^(3*k+1)/Product_{j=k..2*k+1} (1-x^j). - Seiichi Manyama, May 17 2023

Example

a(8) = 4 counts these partitions:  3311, 3221, 32111, 311111.

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], _?(2*Min[#]+1==Max[#]&)], {n, 60}] (* Harvey P. Dale, Jun 25 2017 *)
kmax = 65;
Sum[x^(3k+1)/Product[1-x^j, {j, k, 2k+1}], {k, 1, kmax}]/x + O[x]^kmax // CoefficientList[#, x]& (* Jean-François Alcover, May 30 2024, after Seiichi Manyama *)

Prog

(PARI) my(N=70, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(3*k+1)/prod(j=k, 2*k+1, 1-x^j)))) \\ Seiichi Manyama, May 17 2023

Crossrefs

Cf. A118096, A237829.

Cf. A049820, A237757, A237825, A237826, A237827, A000041.

Sequence in context: A210948 A359678 A008133 * A362607 A340626 A351489

Adjacent sequences: A237825 A237826 A237827 * A237829 A237830 A237831

Keyword

nonn,easy

Author

Clark Kimberling, Feb 16 2014

Status

approved