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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 20 02:20 EDT 2019. Contains 327207 sequences. (Running on oeis4.)