A124227 Number of partitions of n with even crank.

1, 0, 2, 1, 5, 1, 7, 5, 14, 10, 26, 24, 45, 43, 75, 80, 127, 135, 205, 230, 331, 376, 522, 605, 815, 946, 1252, 1470, 1902, 2235, 2852, 3366, 4237, 5001, 6230, 7361, 9081, 10715, 13115, 15475, 18802, 22145, 26742, 31463, 37775, 44362, 52998, 62142

Offset: 0

Author

Vladeta Jovovic, Oct 20 2006

Keywords

• easy
• nonn

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).

Crossrefs

Cf. A000041, A124226, A124228.

Programs

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: A124227 := proc(n) (A000041(n)+A124226(n))/2 ; end: for n from 0 to 60 do printf("%a, ",A124227(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 == 1, 0, (PartitionsP[n] + A132970[n])/2];
Table[a[n], {n, 0, 47}] (* Jean-François Alcover, Oct 26 2023, after Michael Somos in A124226 *)

Formula

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

Extensions

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