A096401 Number of balanced partitions of n into distinct parts: least part is equal to number of parts.

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, 23, 26, 28, 32, 35, 39, 43, 48, 53, 59, 65, 72, 80, 88, 97, 107, 118, 129, 142, 155, 171, 186, 204, 222, 244, 265, 290, 315, 345, 374, 409, 443, 484, 524, 571, 618, 673, 727, 790

Offset: 1

Author

Vladeta Jovovic, Aug 06 2004

Keywords

• easy
• nonn

Examples

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..1000

Crossrefs

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

Cf. A006141, A047993, A339446.

Programs

Maple

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

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

Formula

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

a(n) = A025157(n) - A237979(n) = A237977(n) - A237976(n) for n > 0. - Seiichi Manyama, Jan 13 2022

a(n) ~ (1 - A263719) * A025157(n). - Vaclav Kotesovec, Jan 15 2022

Extensions

More terms from Emeric Deutsch, Mar 29 2005