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Number of partitions of n in which number of least parts is equal to least part.
5

%I #19 Mar 01 2021 11:53:07

%S 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 393,470,584,705,865,1036,1270,1514,1838,2192,2639,3137,3767,4455,

%U 5321,6287,7469,8794,10419,12230,14427,16904,19863,23210,27207,31701,37039

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

%H Alois P. Heinz, <a href="/A096403/b096403.txt">Table of n, a(n) for n = 1..10000</a>

%F G.f.: Sum((x^(m^2)-x^(m*(m+1)))/Product(1-x^i, i=m..infinity), m=1..infinity).

%F G.f.: sum(n>=1, x^(n^2) / prod(k>=n+1,1-x^k)). [_Joerg Arndt_, Mar 23 2011]

%e a(7)=5 because we have 61, 421, 331, 322 and 2221.

%p 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 # second Maple program:

%p b:= proc(n, i) option remember; `if`(n=0, 1,

%p `if`(i>n, 0, b(n, i+1)+b(n-i, i)))

%p end:

%p a:= n-> add(b(n-j^2, j+1), j=1..isqrt(n)):

%p seq(a(n), n=1..55); # _Alois P. Heinz_, Jan 25 2021

%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[i>n, 0, b[n, i+1] + b[n-i, i]]];

%t a[n_] := Sum[b[n - j^2, j+1], {j, 1, Sqrt[n]}];

%t Array[a, 55] (* _Jean-François Alcover_, Mar 01 2021, after _Alois P. Heinz_ *)

%Y Cf. A006141.

%K easy,nonn

%O 1,4

%A _Vladeta Jovovic_, Aug 07 2004

%E More terms from _Emeric Deutsch_, Jul 25 2005