cp's OEIS Frontend

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

%I A237979 #21 Jan 18 2022 06:49:18
%S A237979 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,
%T A237979 40,45,50,56,63,70,78,87,96,107,118,131,144,160,175,194,213,235,257,
%U A237979 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
%N A237979 Number of strict partitions of n such that (least part) > number of parts.
%C A237979 Also the number of partitions into distinct parts with minimal part >= 2 and difference between parts >= 3. [_Joerg Arndt_, Mar 31 2014]
%H A237979 Vaclav Kotesovec, <a href="/A237979/b237979.txt">Table of n, a(n) for n = 1..1000</a>
%F A237979 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]
%F A237979 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
%F A237979 a(n) ~ A263719 * A025157(n). - _Vaclav Kotesovec_, Jan 15 2022
%e A237979 a(9) = 3 counts these partitions:  9, 63, 54.
%t A237979 z = 50; q[n_] := q[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
%t A237979 p1[p_] := p1[p] = DeleteDuplicates[p]; t[p_] := t[p] = Length[p1[p]]
%t A237979 Table[Count[q[n], p_ /; Min[p] < t[p]], {n, z}]  (* A237976 *)
%t A237979 Table[Count[q[n], p_ /; Min[p] <= t[p]], {n, z}] (* A237977 *)
%t A237979 Table[Count[q[n], p_ /; Min[p] == t[p]], {n, z}] (* A096401 *)
%t A237979 Table[Count[q[n], p_ /; Min[p] > t[p]], {n, z}]  (* A237979 *)
%t A237979 Table[Count[q[n], p_ /; Min[p] >= t[p]], {n, z}] (* A025157 *)
%o A237979 (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
%Y A237979 Cf.  A000009, A237976, A096401, A237977, A025157.
%K A237979 nonn,easy
%O A237979 1,7
%A A237979 _Clark Kimberling_, Feb 18 2014