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

1,4

Links

Table of n, a(n) for n=1..40.

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

Example

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.

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);

Crossrefs

Cf. A116687.

Sequence in context: A294425 A123325 A239206 * A293423 A011755 A262466

Adjacent sequences: A116685 A116686 A116687 * A116689 A116690 A116691

Keyword

nonn

Author

Emeric Deutsch, Feb 23 2006

Status

approved