OFFSET
0,56
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).
n-th row contains 2n+1 terms.
LINKS
G. E. Andrews and F. Garvan, Dyson's crank of a partition, Bull. Amer. Math. Soc., 18 (1988), 167-171.
F. Garvan, New combinatorial interpretations of Ramanujan's partition congruences mod 5, 7 and 11, Trans. Amer. Math. Soc., 305 (1988), 47-77.
FORMULA
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
EXAMPLE
{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}.
From Omar E. Pol, Mar 04 2012: (Start)
Triangle begins:
. 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, 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, 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;
(End)
MATHEMATICA
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 *)
PROG
(Sage)
for n in (0..9): # computes the sequence as a triangle
a = [p.crank() for p in Partitions(n)]
[a.count(k) for k in (-n..n)] # Peter Luschny, Sep 15 2014
CROSSREFS
KEYWORD
nonn,tabf,nice,easy
AUTHOR
N. J. A. Sloane, Sep 29 2001
EXTENSIONS
More terms from Vladeta Jovovic, Sep 29 2001
STATUS
approved