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A124228 Number of partitions of n with odd crank.

%I A124228 #14 Oct 26 2023 09:49:55
%S A124228 0,1,0,2,0,6,4,10,8,20,16,32,32,58,60,96,104,162,180,260,296,416,480,
%T A124228 650,760,1012,1184,1540,1816,2330,2752,3476,4112,5142,6080,7522,8896,
%U A124228 10922,12900,15710,18536,22438,26432,31798,37400,44772,52560,62612
%N A124228 Number of partitions of n with odd crank.
%C A124228 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).
%F A124228 a(n) = (A000041(n)-A124226(n))/2.
%p A124228 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
%t A124228 A132970[n_] := SeriesCoefficient[EllipticTheta[4, 0, x] QPochhammer[x, x^2], {x, 0, n}];
%t A124228 a[n_] := If[n < 2, n, (PartitionsP[n] - A132970[n])/2];
%t A124228 Table[a[n], {n, 0, 47}] (* _Jean-François Alcover_, Oct 26 2023, after _Michael Somos_ in A124226 *)
%Y A124228 Cf. A000041, A124226, A124227.
%K A124228 easy,nonn
%O A124228 0,4
%A A124228 _Vladeta Jovovic_, Oct 20 2006
%E A124228 More terms from _R. J. Mathar_, May 18 2007