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

1,4

Comments

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

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

Links

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

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).

Example

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.

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<n, b(n-i, i-1), 0), 0)
    end:
g:= proc(n, i) option remember;
      `if`(i=n, n, 0)+`if`(i<n, g(n, i+1)+g(n-i, i+1), 0)
    end:
a:= n-> 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[i<n, b[n-i, i-1], 0], 0]; g[n_, i_] := g[n, i] = If[i==n, n, 0] + If[i<n, g[n, i+1] + g[n-i, i+1], 0]; a[n_] := g[n, 1] - b[n, n]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Mar 24 2015, after Alois P. Heinz *)

Crossrefs

Cf. A005895, A092265, A117454.

Sequence in context: A192078 A136184 A011908 * A277753 A237820 A110571

Adjacent sequences: A117452 A117453 A117454 * A117456 A117457 A117458

Keyword

nonn

Author

Emeric Deutsch, Mar 18 2006

Status

approved