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%I #41 Sep 16 2014 05:48:04
%S 1,1,0,0,1,0,0,0,1,1,0,0,1,0,0,1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,1,1,0,1,
%T 0,1,1,0,1,1,1,1,1,1,1,1,1,0,1,1,0,1,1,1,1,2,1,2,1,1,1,1,0,1,1,0,1,1,
%U 2,1,2,2,2,2,2,1,2,1,1,0,1,1,0,1,1,2,1,3,2,3,2,3,2,3,1,2,1,1,0,1,1,0,1,1,2
%N Triangle T(n,k) with zeroth row {1} and row n for n >= 1 giving number of partitions of n with crank k, for -n <= k <= n.
%C 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).
%C n-th row contains 2n+1 terms.
%H G. E. Andrews and F. Garvan, <a href="http://dx.doi.org/10.1090/S0273-0979-1988-15637-6">Dyson's crank of a partition</a>, Bull. Amer. Math. Soc., 18 (1988), 167-171.
%H F. Garvan, <a href="http://dx.doi.org/10.1090/S0002-9947-1988-0920146-8">New combinatorial interpretations of Ramanujan's partition congruences mod 5, 7 and 11</a>, Trans. Amer. Math. Soc., 305 (1988), 47-77.
%F G.f. for k-th column is Sum(m>=1, (-1)^m*x^(k*m)*(x^((m^2+m)/2)-x^((m^2-m)/2)))/Product(m>=1, 1-x^m). - _Vladeta Jovovic_, Dec 22 2004
%e {T(20, k), -20 <= k <=20} = {1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 13, 19, 20, 26, 28, 34, 34, 39, 38, 41, 38, 39, 34, 34, 28, 26, 20, 19, 13, 12, 8, 7, 4, 4, 2, 2, 1, 1, 0, 1}.
%e From _Omar E. Pol_, Mar 04 2012: (Start)
%e Triangle begins:
%e . 1;
%e . 1, 0, 0;
%e . 1, 0, 0, 0, 1;
%e . 1, 0, 0, 1, 0, 0, 1;
%e . 1, 0, 1, 0, 1, 0, 1, 0, 1;
%e . 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1;
%e . 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1;
%e . 1, 0, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 0, 1;
%e . 1, 0, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 1, 0, 1;
%e 1, 0, 1, 1, 2, 1, 3, 2, 3, 2, 3, 2, 3, 1, 2, 1, 1, 0, 1;
%e (End)
%t max = 12; f[k_ /; k < 0] := f[-k]; f[k_] := Sum[(-1)^m*x^(k*m)*(x^((m^2 + m)/2) - x^((m^2 - m)/2)), {m, 1, max}]/Product[1 - x^m, {m, 1, max}]; t = Table[ Series[f[k], {x, 0, max}] // CoefficientList[#, x]&, {k, -(max-2), max-2}] // Transpose; Table[If[n == 2, {1, 0, 0}, Table[t[[n, k]], {k, max-n, max+n-2}]], {n, 1, max-1}] // Flatten (* _Jean-François Alcover_, Apr 11 2013, after _Vladeta Jovovic_ *)
%o (Sage)
%o for n in (0..9): # computes the sequence as a triangle
%o a = [p.crank() for p in Partitions(n)]
%o [a.count(k) for k in (-n..n)] # _Peter Luschny_, Sep 15 2014
%Y Cf. A001522, A064410, A064428.
%Y Row sums give A000041. - _Omar E. Pol_, Mar 04 2012
%K nonn,tabf,nice,easy
%O 0,56
%A _N. J. A. Sloane_, Sep 29 2001
%E More terms from _Vladeta Jovovic_, Sep 29 2001
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