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

0,3

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: 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 *)

Crossrefs

Cf. A000041, A124226, A124228.

Sequence in context: A260147 A263454 A036073 * A064865 A178472 A337667

Adjacent sequences: A124224 A124225 A124226 * A124228 A124229 A124230

Keyword

easy,nonn

Author

Vladeta Jovovic, Oct 20 2006

Extensions

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

Status

approved