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%I #37 Jan 21 2020 00:28:36
%S 0,2,8,20,42,80,140,238,380,602,910,1372,1996,2900,4102,5790,8002,
%T 11046,14980,20282,27090,36092,47546,62510,81374,105700,136210,175084,
%U 223510,284694,360410,455244,572054,717160,894964,1114470,1382032,1710262,2108750,2594704,3182120
%N The second rank moment function N_2(n).
%C N_2(n) is also called the second Atkin-Garvan moment (see Andrews' paper). - _Omar E. Pol_, Oct 23 2013
%H Alois P. Heinz, <a href="/A220908/b220908.txt">Table of n, a(n) for n = 1..1000</a>
%H G. E. Andrews, <a href="http://www.math.psu.edu/vstein/alg/antheory/preprint/andrews/17.pdf">The number of smallest parts in the partitions of n</a>, p. 3
%H F. G. Garvan, <a href="http://www.math.ufl.edu/~fgarvan/papers/hspt.pdf">Higher-order spt functions</a>, Adv. Math. 228 (2011), no. 1, 241-265. See Eq. (1.1).
%F a(n) = 2*A220907(n) = 2*(n*A000041(n)-A092269(n)).
%F a(n) = 2*(A066186(n) - A092269(n)). - _Omar E. Pol_, Jan 09 2013
%F a(n) = A220909(n) - A211982(n). - _Omar E. Pol_, Jan 16 2013
%F a(n) ~ exp(Pi*sqrt(2*n/3))/(2*sqrt(3)) * (1 - (3*sqrt(6)/(2*Pi) + Pi/(24*sqrt(6)))/sqrt(n) + (5/48 + Pi^2/6912)/n). - _Vaclav Kotesovec_, Jul 31 2017
%p b:= proc(n, i) option remember; `if`(n=0 or i<1, 0,
%p `if`(irem(n, i, 'r')=0, r, 0)+add(b(n-i*j, i-1), j=0..n/i))
%p end:
%p a:= n-> 2*(n*combinat[numbpart](n)- b(n, n)):
%p seq(a(n), n=1..60); # _Alois P. Heinz_, Jan 09 2013
%t terms = 41; gf = Sum[x^n/(1 - x^n)*Product[1/(1 - x^k), {k, n, terms}], {n, 1, terms}]; spt = CoefficientList[ Series[gf, {x, 0, terms}], x] // Rest; a[n_] := 2*(n*PartitionsP[n] - spt[[n]]); Table[a[n], {n, 1, terms}] (* _Jean-François Alcover_, Jan 17 2013, after g.f. of spt(n) *)
%Y Cf. A000041, A092269, A220907.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Jan 02 2013
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