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
Keywords
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
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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).