A220909 The second crank moment function M_2(n).

0, 2, 8, 18, 40, 70, 132, 210, 352, 540, 840, 1232, 1848, 2626, 3780, 5280, 7392, 10098, 13860, 18620, 25080, 33264, 44088, 57730, 75600, 97900, 126672, 162540, 208208, 264770, 336240, 424204, 534336, 669438, 837080, 1041810, 1294344, 1601138, 1977140, 2432430, 2987040, 3655806

Offset: 0

Author

N. J. A. Sloane, Jan 02 2013

Keywords

• nonn

Comments

M_2(n) is defined to be Sum_{m=-n..n} m^2 M(m,n) where M(m,n) is the number of partitions of n with crank m except for n=1 where M(-1,1) = M(1,1) = -M(0,1) = 1. - Michael Somos, Nov 10 2013

From Omar E. Pol, Jul 25 2022: (Start)

Apart from the initial zero this is also:

Convolution of A074400 and A000041.

Convolution of A000203 and A139582. (End)

Examples

			G.f. = 2*x + 8*x^2 + 18*x^3 + 40*x^4 + 70*x^5 + 132*x^6 + 210*x^7 + ...
For n=1, M_2(1) = Sum_{m=-1..1} m^2 * M(m,2) = (-1)^2*1 + 0^2*(-1) + 1^2*1 = 2. For n=2, the partition [2] has crank 2 and partition [1,1] has crank -2, hence M_2(2) = 2^2 + (-2)^2 = 8. - _Michael Somos_, Nov 10 2013
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• F. G. Garvan, Higher-order spt functions, arXiv:1008.1207 [math.NT], 2010.
• F. G. Garvan, Higher-order spt functions, Adv. Math. 228 (2011), no. 1, 241-265.
• Wikipedia, Crank of a partition

Crossrefs

Cf. A000041, A066186, A092269, A220908.

Cf. A000203, A074400, A139582.

Programs

Mathematica

a[ n_] := 2 n PartitionsP @ n (* Michael Somos, Nov 10 2013 *)

PARI

{a(n) = if( n<0, 0, 2 * n * polcoeff( 1 / eta(x + x * O(x^n)), n))} /* Michael Somos, Nov 10 2013 */

Formula

a(n) = 2*n*A000041(n) = 2*A066186(n).

a(n) = n*A139582(n). - Omar E. Pol, Jan 03 2013

a(n) = A220908(n) + A211982(n), n >= 1. - Omar E. Pol, Jan 17 2013

a(n) = 2*(A092269(n) + A220907(n)), n >= 1. _Omar E. Pol, Feb 18 2013

a(n) ~ exp(Pi*sqrt(2*n/3))/(2*sqrt(3)) * (1 - (sqrt(3/2)/Pi + Pi/(24*sqrt(6))) / sqrt(n)). - Vaclav Kotesovec, Oct 24 2016