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A096403 Number of partitions of n in which number of least parts is equal to least part.

%I A096403 #21 Jun 15 2025 03:19:52
%S A096403 1,0,1,2,2,2,5,5,9,10,14,17,26,30,41,52,67,81,108,129,168,204,257,311,
%T A096403 393,470,584,705,865,1036,1270,1514,1838,2192,2639,3137,3767,4455,
%U A096403 5321,6287,7469,8794,10419,12230,14427,16904,19863,23210,27207,31701,37039
%N A096403 Number of partitions of n in which number of least parts is equal to least part.
%H A096403 Alois P. Heinz, <a href="/A096403/b096403.txt">Table of n, a(n) for n = 1..10000</a>
%F A096403 G.f.: Sum((x^(m^2)-x^(m*(m+1)))/Product(1-x^i, i=m..infinity), m=1..infinity).
%F A096403 G.f.: sum(n>=1, x^(n^2) / prod(k>=n+1,1-x^k)). [_Joerg Arndt_, Mar 23 2011]
%F A096403 a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(5/2) * n^(3/2)). - _Vaclav Kotesovec_, Jun 15 2025
%e A096403 a(7)=5 because we have 61, 421, 331, 322 and 2221.
%p A096403 G:=sum((x^(m^2)-x^(m*(m+1)))/product(1-x^i,i=m..80),m=1..80): Gser:=series(G,x=0,70): seq(coeff(Gser,x^n),n=1..60); # _Emeric Deutsch_, Jul 25 2005
%p A096403 # second Maple program:
%p A096403 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A096403      `if`(i>n, 0, b(n, i+1)+b(n-i, i)))
%p A096403     end:
%p A096403 a:= n-> add(b(n-j^2, j+1), j=1..isqrt(n)):
%p A096403 seq(a(n), n=1..55);  # _Alois P. Heinz_, Jan 25 2021
%t A096403 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i>n, 0, b[n, i+1] + b[n-i, i]]];
%t A096403 a[n_] := Sum[b[n - j^2, j+1], {j, 1, Sqrt[n]}];
%t A096403 Array[a, 55] (* _Jean-François Alcover_, Mar 01 2021, after _Alois P. Heinz_ *)
%Y A096403 Cf. A006141.
%K A096403 easy,nonn
%O A096403 1,4
%A A096403 _Vladeta Jovovic_, Aug 07 2004
%E A096403 More terms from _Emeric Deutsch_, Jul 25 2005