cp's OEIS Frontend

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

%I A237757 #17 Jan 22 2022 07:31:08
%S A237757 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,
%T A237757 86,100,114,133,151,174,198,227,257,295,333,379,428,486,547,620,696,
%U A237757 786,882,993,1111,1250,1396,1565,1747,1954,2176,2431,2703,3013
%N A237757 Number of partitions of n such that 2*(least part) = (number of parts).
%H A237757 Robert Israel, <a href="/A237757/b237757.txt">Table of n, a(n) for n = 1..10000</a>
%F A237757 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
%F A237757 a(n) ~ exp(Pi*sqrt(n/3)) / (2^(7/2) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jan 22 2022
%e A237757 a(8) = 2 counts these partitions: 71, 2222.
%p A237757 f:= proc(n) local t, k, np;
%p A237757   t:= 0;
%p A237757   for k from 1 do
%p A237757     np:= n - 1 - 2*k*(k-1);
%p A237757     if np < 2*k-1 then return t fi;
%p A237757     t:= t + combinat:-numbpart(np, 2*k-1) - combinat:-numbpart(np,2*k-2)
%p A237757   od;
%p A237757 end proc:
%p A237757 map(f, [$1..100]); # _Robert Israel_, Jul 01 2020
%t A237757 z = 50; Table[Count[IntegerPartitions[n], p_ /; 2 Min[p] == Length[p]], {n, z}]
%Y A237757 Cf. A237753.
%K A237757 nonn,easy
%O A237757 1,8
%A A237757 _Clark Kimberling_, Feb 13 2014