A237976 Number of strict partitions of n such that (least part) < number of parts.

0, 0, 0, 1, 1, 1, 2, 2, 3, 4, 6, 7, 9, 11, 14, 17, 21, 25, 31, 37, 45, 54, 64, 76, 90, 106, 124, 146, 170, 198, 230, 267, 308, 357, 410, 472, 542, 621, 709, 811, 923, 1051, 1194, 1355, 1534, 1738, 1962, 2215, 2497, 2812, 3161, 3553, 3986, 4469, 5005, 5600, 6258

Offset

0,7

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*(k-1))) / Product_{j=1..k} (1-x^j). - Seiichi Manyama, Jan 13 2022

a(n) = A000009(n) - A025157(n). - Vaclav Kotesovec, Jan 18 2022

Example

a(8) = 3 counts these partitions:  71, 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, 0, 0], Vec(sum(k=0, N, x^(k*(k+1)/2)*(1-x^(k*(k-1)))/prod(j=1, k, 1-x^j)))) \\ Seiichi Manyama, Jan 13 2022

Crossrefs

Cf. A000009, A237977, A096401, A237979, A025157, A039899.

Sequence in context: A067835 A029011 A077117 * A035365 A335745 A119604

Adjacent sequences: A237973 A237974 A237975 * A237977 A237978 A237979

Keyword

nonn,easy

Author

Clark Kimberling, Feb 18 2014

Extensions

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

Status

approved