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

Author

Emeric Deutsch, Feb 23 2006

Keywords

• nonn

Comments

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

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

Examples

			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).
		

Links

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

Crossrefs

Cf. A116685, A211870, A211881.

Programs

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 *)

Formula

a(n) = Sum_{k>=0} k*A116685(n,k).

G.f.: Sum_{i>=1} (x^i*(Sum_{j=1..i-1} x^j/(1-x^j))/(Product_{q=1..i} (1-x^q))).

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

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