A237977 Number of strict partitions of n such that (least part) <= number of parts.

0, 1, 0, 1, 1, 2, 3, 3, 4, 5, 7, 8, 11, 13, 17, 20, 25, 29, 36, 42, 51, 60, 72, 84, 100, 117, 137, 160, 187, 216, 251, 290, 334, 385, 442, 507, 581, 664, 757, 864, 982, 1116, 1266, 1435, 1622, 1835, 2069, 2333, 2626, 2954, 3316, 3724, 4172, 4673, 5227, 5844

Offset

0,6

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

G.f.: Sum_{k>=0} x^(k*(k+1)/2) * (1-x^(k^2)) / Product_{j=1..k} (1-x^j). - Seiichi Manyama, Jan 13 2022

a(n) = A000009(n) - A237979(n). - Vaclav Kotesovec, Jan 18 2022

Example

a(8) = 4 counts these partitions:  71, 53, 521, 431.

Mathematica

z = 50; q[n_] := q[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
p1[p_] := p1[p] = DeleteDuplicates[p]; t[p_] := t[p] = Length[p1[p]]
Table[Count[q[n], p_ /; Min[p] < t[p]], {n, z}]  (* A237976 *)
Table[Count[q[n], p_ /; Min[p] <= t[p]], {n, z}] (* A237977 *)
Table[Count[q[n], p_ /; Min[p] == t[p]], {n, z}] (* A096401 *)
Table[Count[q[n], p_ /; Min[p] > t[p]], {n, z}]  (* A237979 *)
Table[Count[q[n], p_ /; Min[p] >= t[p]], {n, z}] (* A025157 *)

Prog

(PARI) my(N=66, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, x^(k*(k+1)/2)*(1-x^k^2)/prod(j=1, k, 1-x^j)))) \\ Seiichi Manyama, Jan 13 2022

Crossrefs

Cf. A000009, A237976, A096401, A237979, A025157, A039900.

Sequence in context: A029034 A343941 A280127 * A115339 A305631 A036019

Adjacent sequences: A237974 A237975 A237976 * A237978 A237979 A237980

Keyword

nonn,easy

Author

Clark Kimberling, Feb 18 2014

Extensions

Prepended a(0)=0, Seiichi Manyama, Jan 13 2022

Status

approved