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A124227 Number of partitions of n with even crank.

%I A124227 #15 Oct 26 2023 09:50:16
%S A124227 1,0,2,1,5,1,7,5,14,10,26,24,45,43,75,80,127,135,205,230,331,376,522,
%T A124227 605,815,946,1252,1470,1902,2235,2852,3366,4237,5001,6230,7361,9081,
%U A124227 10715,13115,15475,18802,22145,26742,31463,37775,44362,52998,62142
%N A124227 Number of partitions of n with even crank.
%C A124227 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 A124227 a(n) = (A000041(n) + A124226(n))/2.
%p A124227 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
%t A124227 A132970[n_] := SeriesCoefficient[EllipticTheta[4, 0, x] QPochhammer[x, x^2], {x, 0, n}];
%t A124227 a[n_] := If[n == 1, 0, (PartitionsP[n] + A132970[n])/2];
%t A124227 Table[a[n], {n, 0, 47}] (* _Jean-François Alcover_, Oct 26 2023, after _Michael Somos_ in A124226 *)
%Y A124227 Cf. A000041, A124226, A124228.
%K A124227 easy,nonn
%O A124227 0,3
%A A124227 _Vladeta Jovovic_, Oct 20 2006
%E A124227 More terms from _R. J. Mathar_, May 18 2007