A117455 Sum of the differences between the largest part and smallest part over all partitions of n into distinct parts.

0, 0, 1, 2, 4, 8, 12, 19, 27, 41, 54, 76, 99, 133, 171, 223, 279, 357, 443, 554, 682, 841, 1022, 1247, 1504, 1814, 2174, 2603, 3092, 3676, 4346, 5127, 6030, 7076, 8275, 9669, 11254, 13078, 15167, 17556, 20270, 23377, 26899, 30902, 35448, 40592, 46403

Offset: 1

Author

Emeric Deutsch, Mar 18 2006

Keywords

• nonn

Comments

a(n) = sum(k*A117454(n,k), k=0..n-2).

a(n) = A005895(n)-A092265(n). - Alois P. Heinz, Jul 06 2012

Examples

			a(7)=12 because the partitions of 7 into distinct parts are [7], [6,1], [5,2], [4,3] and [4,2,1] and (7-7)+(6-1)+(5-2)+(4-3)+(4-1)=12.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005895, A092265, A117454.

Programs

Maple

g:=sum(x^(i*(i+1)/2)*sum(1/(1-x^j),j=1..i-1)/product(1-x^j,j=1..i),i=1..15): gser:=series(g,x=0,55): seq(coeff(gser,x^n), n=1..50);
# second Maple program:
b:= proc(n, i) option remember;
      `if`(i=n, n, 0)+`if`(i>0, b(n, i-1)+
      `if`(i g(n, 1) -b(n, n):
seq(a(n), n=1..60);  # Alois P. Heinz, Jul 06 2012

Mathematica

b[n_, i_] := b[n, i] = If[i==n, n, 0] + If[i>0, b[n, i-1] + If[iJean-François Alcover, Mar 24 2015, after Alois P. Heinz *)

Formula

G.f.: sum(x^(i(i+1)/2)*sum(1/(1-x^j), j=1..i-1)/product(1-x^j, j=1..i), i=1..infinity) (obtained by taking the derivative with respect to t of the g.f. G(t,x) of A117454 and letting t=1).