A117471 The difference between the largest part and the smallest part summed over all those partitions of n in which every integer from the smallest part to the largest part occurs.

0, 0, 1, 1, 3, 4, 7, 8, 14, 17, 24, 30, 40, 50, 67, 79, 101, 126, 153, 186, 231, 276, 332, 399, 477, 567, 677, 795, 938, 1111, 1294, 1512, 1773, 2058, 2392, 2775, 3204, 3701, 4272, 4904, 5630, 6467, 7387, 8442, 9651, 10980, 12491, 14202, 16109, 18260, 20680

Offset

1,5

Comments

a(n)=sum(k*A117470(n,k),k>=0).

Links

Table of n, a(n) for n=1..51.

Formula

G.f.=sum(x^j*product(1+x^i, i=1..j-1)sum(x^i/(1+x^i), i=1..j-1)/(1-x^j), j=1..infinity) (obtained by taking the derivative with respect to t of the g.f. G(t,x) of A117470 and setting t=1).

Example

a(6)=4 because the 7 (=A034296(6) ) partitions of 6 in which every integer from the smallest part to the largest part occurs are [6],[3,3],[3,2,1],[2,2,2],[2,2,1,1],[2,1,1,1,1],[1,1,1,1,1,1] and (6-6)+(3-3)+(3-1)+(2-2)+(2-1)+(2-1)+(1-1)=4.

Maple

g:=sum(x^j*product(1+x^i, i=1..j-1)*sum(x^i/(1+x^i), i=1..j-1)/(1-x^j), j=1..65): gser:=series(g, x=0, 60): seq(coeff(gser, x, n), n=1..57);

Mathematica

Table[Total[Max[#]-Min[#]&/@Select[IntegerPartitions[n], Max[Abs[ Differences[ #]]]<2&]], {n, 60}] (* Harvey P. Dale, Oct 14 2014 *)

Crossrefs

Cf. A034296, A117470.

Sequence in context: A120355 A114210 A073271 * A217135 A323247 A285662

Adjacent sequences: A117468 A117469 A117470 * A117472 A117473 A117474

Keyword

nonn

Author

Emeric Deutsch, Mar 20 2006

Status

approved