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

1,8

Links

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

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

Example

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

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}]

Crossrefs

Cf. A237753.

Sequence in context: A286316 A027198 A238215 * A027197 A363221 A332577

Adjacent sequences: A237754 A237755 A237756 * A237758 A237759 A237760

Keyword

nonn,easy

Author

Clark Kimberling, Feb 13 2014

Status

approved