A116688 Sum over all partitions of n of the sum of the parts that are smaller than the largest part.

0, 0, 1, 3, 9, 17, 36, 61, 106, 171, 273, 411, 627, 916, 1326, 1890, 2667, 3698, 5102, 6943, 9388, 12588, 16747, 22113, 29051, 37914, 49191, 63515, 81589, 104315, 132799, 168351, 212540, 267395, 335085, 418574, 521093, 646763, 800164, 987315

Offset: 1

Author

Emeric Deutsch, Feb 23 2006

Keywords

• nonn

Examples

			a(5)=9 because the partitions of 5 are [5],[4,(1)],[3,(2)],[3,(1),(1)],
[2,2,(1)],[2,(1),(1),(1)] and [1,1,1,1,1] and the sum of the parts (shown between parentheses) that are smaller than the largest part is 1+2+1+1+1+1+1+1=9.
		

Crossrefs

Cf. A116687.

Programs

Maple

f:=sum(x^i*sum(j*x^j/(1-x^j),j=1..i-1)/product(1-x^q,q=1..i),i=1..55): fser:=series(f,x=0,50): seq(coeff(fser,x^n),n=1..47);

Formula

G.f.=sum(x^i*sum(jx^j/(1-x^j), j=1..i-1)/product(1-x^q, q=1..i), i=1..infinity).

a(n) = Sum_{k>=0} k * A116687(n,k).

a(n) = n*A000041(n) - A092321(n). - Vladeta Jovovic, Feb 24 2006