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A124228
Number of partitions of n with odd crank.
3
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).
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
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Oct 20 2006
EXTENSIONS
More terms from R. J. Mathar, May 18 2007
STATUS
approved