A124228 Number of partitions of n with odd crank.

0, 1, 0, 2, 0, 6, 4, 10, 8, 20, 16, 32, 32, 58, 60, 96, 104, 162, 180, 260, 296, 416, 480, 650, 760, 1012, 1184, 1540, 1816, 2330, 2752, 3476, 4112, 5142, 6080, 7522, 8896, 10922, 12900, 15710, 18536, 22438, 26432, 31798, 37400, 44772, 52560, 62612

Offset

0,4

Comments

For a partition p, let l(p) = largest part of p, w(p) = number of 1's in p, m(p) = number of parts of p larger than w(p). The crank of p is given by l(p) if w(p) = 0, otherwise m(p)-w(p).

Links

Table of n, a(n) for n=0..47.

Formula

a(n) = (A000041(n)-A124226(n))/2.

Maple

A000041 := proc(n) combinat[numbpart](n) ; end: A124226 := proc(n) local x, gf, i ; gf := 1; for i from 1 to n+1 do gf := taylor(gf*(1-x^i)/(1+x^i)^2, x=0, n+1) ; od ; coeftayl(2*x+gf, x=0, n) ; end: A124228 := proc(n) (A000041(n)-A124226(n))/2 ; end: for n from 0 to 60 do printf("%a, ", A124228(n)) ; od ; # R. J. Mathar, May 18 2007

Mathematica

A132970[n_] := SeriesCoefficient[EllipticTheta[4, 0, x] QPochhammer[x, x^2], {x, 0, n}];
a[n_] := If[n < 2, n, (PartitionsP[n] - A132970[n])/2];
Table[a[n], {n, 0, 47}] (* Jean-François Alcover, Oct 26 2023, after Michael Somos in A124226 *)

Crossrefs

Cf. A000041, A124226, A124227.

Sequence in context: A242561 A131595 A290826 * A115879 A115880 A335728

Adjacent sequences: A124225 A124226 A124227 * A124229 A124230 A124231

Keyword

easy,nonn

Author

Vladeta Jovovic, Oct 20 2006

Extensions

More terms from R. J. Mathar, May 18 2007

Status

approved