A237829 Number of partitions of n such that 2*(least part) - 1 = greatest part.

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

Offset

1,5

Links

Table of n, a(n) for n=1..64.

Formula

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

Example

a(8) = 3 counts these partitions:  53, 332, 11111111.

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 *)
(* Second program: *)
kmax = 64;
Sum[x^(3k-1)/Product[1-x^j, {j, k, 2k-1}], {k, 1, kmax}]/x+1+O[x]^kmax // CoefficientList[#, x]& (* Jean-François Alcover, May 30 2024, after Seiichi Manyama *)

Prog

(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

Crossrefs

Cf. A118096, A237828.

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

Sequence in context: A112218 A172366 A132148 * A159974 A143866 A155002

Adjacent sequences: A237826 A237827 A237828 * A237830 A237831 A237832

Keyword

nonn,easy

Author

Clark Kimberling, Feb 16 2014

Status

approved