A237979 Number of strict partitions of n such that (least part) > number of parts.

0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 7, 7, 9, 10, 12, 13, 16, 17, 20, 22, 25, 28, 32, 35, 40, 45, 50, 56, 63, 70, 78, 87, 96, 107, 118, 131, 144, 160, 175, 194, 213, 235, 257, 284, 310, 342, 373, 410, 447, 491, 534, 585, 637, 696, 756, 826, 896, 977, 1060, 1153, 1250, 1359, 1471, 1597, 1729, 1874, 2026, 2195, 2371, 2565

Offset: 1

Author

Clark Kimberling, Feb 18 2014

Keywords

• nonn
• easy

Comments

Also the number of partitions into distinct parts with minimal part >= 2 and difference between parts >= 3. [Joerg Arndt, Mar 31 2014]

Examples

			a(9) = 3 counts these partitions:  9, 63, 54.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000009, A237976, A096401, A237977, A025157.

Programs

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 *)

PARI

N=66; q='q+O('q^N); Vec(-1+sum(n=0, N, q^(n*(3*n+1)/2) / prod(k=1, n, 1-q^k ) )) \\ Joerg Arndt, Mar 09 2014

Formula

G.f. with a(0)=0: sum(n>=0, q^(n*(3*n+1)/2) / prod(k=1..n, 1-q^k ) ). [Joerg Arndt, Mar 09 2014]

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(Pi*(1 + 3*r^2)) * n^(3/4)), where r = A263719 and c = 3*(log(r))^2/2 + polylog(2, 1-r). - Vaclav Kotesovec, Jan 15 2022

a(n) ~ A263719 * A025157(n). - Vaclav Kotesovec, Jan 15 2022