A237757 Number of partitions of n such that 2*(least part) = (number of parts).

0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 13, 16, 18, 22, 25, 30, 35, 41, 47, 56, 64, 75, 86, 100, 114, 133, 151, 174, 198, 227, 257, 295, 333, 379, 428, 486, 547, 620, 696, 786, 882, 993, 1111, 1250, 1396, 1565, 1747, 1954, 2176, 2431, 2703, 3013

Offset: 1

Author

Clark Kimberling, Feb 13 2014

Keywords

• nonn
• easy

Examples

			a(8) = 2 counts these partitions: 71, 2222.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A237753.

Programs

Maple

f:= proc(n) local t, k, np;
  t:= 0;
  for k from 1 do
    np:= n - 1 - 2*k*(k-1);
    if np < 2*k-1 then return t fi;
    t:= t + combinat:-numbpart(np, 2*k-1) - combinat:-numbpart(np,2*k-2)
  od;
end proc:
map(f, [$1..100]); # Robert Israel, Jul 01 2020

Mathematica

z = 50; Table[Count[IntegerPartitions[n], p_ /; 2 Min[p] == Length[p]], {n, z}]

Formula

Conjectural g.f.: Sum_{n >= 0} q^(2*(n+1)^2)/Product_{k = 1..2*n+1} 1 - q^k. - Peter Bala, Feb 02 2021

a(n) ~ exp(Pi*sqrt(n/3)) / (2^(7/2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jan 22 2022