A116686 Total number of parts smaller than the largest part, in all partitions of n.

0, 0, 1, 3, 8, 15, 29, 48, 79, 123, 188, 276, 404, 575, 808, 1122, 1540, 2089, 2811, 3748, 4958, 6519, 8504, 11034, 14231, 18268, 23312, 29638, 37486, 47245, 59279, 74140, 92347, 114703, 141933, 175174, 215478, 264407, 323448, 394788, 480509, 583609

Offset

1,4

Comments

a(n) = Sum(k*A116685(n,k), k>=0).

Also, sum over all partitions of n of the difference between the largest part and the smallest part. - Franklin T. Adams-Watters, Feb 29 2008

a(n) = A211870(n) + A211881(n). - Alois P. Heinz, Feb 13 2013

Row sums of A244966. - Omar E. Pol, Jul 19 2014

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Formula

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

a(n) = A006128(n) - A046746(n). - Vladeta Jovovic, Feb 24 2006

Example

a(5) = 8 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], containing a total of 8 parts that are smaller than the largest part (shown between parentheses).

Maple

f:=sum(x^i*sum(x^j/(1-x^j), j=1..i-1)/product(1-x^q, q=1..i), i=2..55): fser:=series(f, x=0, 50): seq(coeff(fser, x^n), n=1..47);

Mathematica

Table[Length[Flatten[Rest[Split[#]]&/@Select[IntegerPartitions[n], #[[1]]> #[[-1]]&]]], {n, 50}] (* Harvey P. Dale, Jul 26 2016 *)

Crossrefs

Cf. A116685, A211870, A211881.

Sequence in context: A294426 A097589 A015631 * A317252 A135350 A068038

Adjacent sequences: A116683 A116684 A116685 * A116687 A116688 A116689

Keyword

nonn

Author

Emeric Deutsch, Feb 23 2006

Status

approved