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The second crank moment function M_2(n).
12

%I #64 Aug 24 2022 09:47:37

%S 0,2,8,18,40,70,132,210,352,540,840,1232,1848,2626,3780,5280,7392,

%T 10098,13860,18620,25080,33264,44088,57730,75600,97900,126672,162540,

%U 208208,264770,336240,424204,534336,669438,837080,1041810,1294344,1601138,1977140,2432430,2987040,3655806

%N The second crank moment function M_2(n).

%C 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

%C From _Omar E. Pol_, Jul 25 2022: (Start)

%C Apart from the initial zero this is also:

%C Convolution of A074400 and A000041.

%C Convolution of A000203 and A139582. (End)

%H G. C. Greubel, <a href="/A220909/b220909.txt">Table of n, a(n) for n = 0..5000</a>

%H F. G. Garvan, <a href="https://arxiv.org/abs/1008.1207">Higher-order spt functions</a>, arXiv:1008.1207 [math.NT], 2010.

%H F. G. Garvan, <a href="https://doi.org/10.1016/j.aim.2011.05.013">Higher-order spt functions</a>, Adv. Math. 228 (2011), no. 1, 241-265.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Crank_of_a_partition">Crank of a partition</a>

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

%F a(n) = n*A139582(n). - _Omar E. Pol_, Jan 03 2013

%F a(n) = A220908(n) + A211982(n), n >= 1. - _Omar E. Pol_, Jan 17 2013

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

%F 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

%e G.f. = 2*x + 8*x^2 + 18*x^3 + 40*x^4 + 70*x^5 + 132*x^6 + 210*x^7 + ...

%e 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

%t a[ n_] := 2 n PartitionsP @ n (* _Michael Somos_, Nov 10 2013 *)

%o (PARI) {a(n) = if( n<0, 0, 2 * n * polcoeff( 1 / eta(x + x * O(x^n)), n))} /* _Michael Somos_, Nov 10 2013 */

%Y Cf. A000041, A066186, A092269, A220908.

%Y Cf. A000203, A074400, A139582.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Jan 02 2013