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A064410 Number of partitions of n with zero crank. 4
0, 0, 1, 1, 1, 1, 1, 2, 2, 4, 4, 7, 7, 11, 12, 17, 19, 27, 30, 41, 48, 62, 73, 95, 110, 140, 166, 206, 243, 302, 354, 435, 513, 622, 733, 887, 1039, 1249, 1467, 1750, 2049, 2438, 2847, 3371, 3934, 4634, 5398, 6343, 7367, 8626, 10009, 11677, 13521, 15737, 18184 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,8

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).

LINKS

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

FORMULA

a(n) = A000041(n)-2*A001522(n). a(n) = A064391(n, 0).

a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi / (3 * 2^(9/2) * n^(3/2)). - Vaclav Kotesovec, May 06 2018

EXAMPLE

a(10)=4 because there are 4 partitions of 10 with zero crank: 1+1+2+3+3, 1+1+4+4, 1+1+3+5 and 1+9.

MATHEMATICA

nmax = 60; Rest[CoefficientList[Series[x - 1 + Sum[(-1)^k*(x^(k*(k + 1)/2) - x^(k*(k - 1)/2)), {k, 1, nmax}] / Product[1 - x^k, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 26 2016 *)

Flatten[{0, Table[PartitionsP[n] - 2*Sum[(-1)^(j+1)*PartitionsP[n - j*((j+1)/2)], {j, 1, Floor[(Sqrt[8*n + 1] - 1)/2]}], {n, 2, 60}]}] (* Vaclav Kotesovec, Sep 26 2016 *)

PROG

(Sage)

[[p.crank() for p in Partitions(n)].count(0) for n in (1..20)] # Peter Luschny, Sep 15 2014

CROSSREFS

Cf. A064391, A000041, A001522.

Sequence in context: A323539 A280954 A197122 * A304178 A266776 A062896

Adjacent sequences:  A064407 A064408 A064409 * A064411 A064412 A064413

KEYWORD

nonn

AUTHOR

Vladeta Jovovic, Sep 29 2001

EXTENSIONS

More terms from Reiner Martin (reinermartin(AT)hotmail.com), Dec 26 2001

STATUS

approved

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Last modified February 25 22:44 EST 2020. Contains 332270 sequences. (Running on oeis4.)