%I #33 Jan 16 2022 11:09:28

%S 1,0,0,0,1,1,1,1,1,1,1,2,2,3,3,4,4,5,5,6,6,8,8,10,11,13,14,17,18,21,

%T 23,26,28,32,35,39,43,48,53,59,65,72,80,88,97,107,118,129,142,155,171,

%U 186,204,222,244,265,290,315,345,374,409,443,484,524,571,618,673,727,790

%N Number of balanced partitions of n into distinct parts: least part is equal to number of parts.

%H Seiichi Manyama, <a href="/A096401/b096401.txt">Table of n, a(n) for n = 1..1000</a>

%F G.f.: Sum_{m>=1} (x^(m*(3*m-1)/2)-x^(m*(3*m+1)/2))/Product_{i=1..m} (1-x^i).

%F a(n) = A025157(n) - A237979(n) = A237977(n) - A237976(n) for n > 0. - _Seiichi Manyama_, Jan 13 2022

%F a(n) ~ (1 - A263719) * A025157(n). - _Vaclav Kotesovec_, Jan 15 2022

%e a(14)=3 because we have 12+2, 7+4+3 and 6+5+3.

%p G:=sum((x^(m*(3*m-1)/2)-x^(m*(3*m+1)/2))/product(1-x^i,i=1..m),m=1..20): Gser:=series(G,x=0,80): seq(coeff(Gser,x^n),n=1..78); # _Emeric Deutsch_, Mar 29 2005

%o (PARI) my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, x^(k*(3*k-1)/2)/prod(j=1, k-1, 1-x^j))) \\ _Seiichi Manyama_, Jan 15 2022

%Y Cf. A000009, A025157, A237976, A237977, A237979.

%Y Cf. A006141, A047993, A339446.

%K easy,nonn

%O 1,12

%A _Vladeta Jovovic_, Aug 06 2004

%E More terms from _Emeric Deutsch_, Mar 29 2005