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A064391 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. 4
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, 1, 0, 1, 1, 2 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..104.

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

Cf. A001522, A064410, A064428.

Row sums give A000041. - Omar E. Pol, Mar 04 2012

Sequence in context: A128583 A218854 A172303 * A236470 A206589 A086011

Adjacent sequences:  A064388 A064389 A064390 * A064392 A064393 A064394

KEYWORD

nonn,tabf,nice,easy

AUTHOR

N. J. A. Sloane, Sep 29 2001

EXTENSIONS

More terms from Vladeta Jovovic, Sep 29 2001

STATUS

approved

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Last modified November 13 11:09 EST 2018. Contains 317133 sequences. (Running on oeis4.)