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%I #31 Jul 31 2017 08:04:02
%S 0,1,4,10,21,40,70,119,190,301,455,686,998,1450,2051,2895,4001,5523,
%T 7490,10141,13545,18046,23773,31255,40687,52850,68105,87542,111755,
%U 142347,180205,227622,286027,358580,447482,557235,691016,855131,1054375,1297352,1591060
%N a(n) = n*p(n)-spt(n) (= n*A000041(n) - A092269(n)).
%H Alois P. Heinz, <a href="/A220907/b220907.txt">Table of n, a(n) for n = 1..1000</a>
%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). - From _N. J. A. Sloane_, Jan 02 2013
%F a(n) = N_2(n)/2 (where N_2(n) is the second rank moment, see relation (1.1) of the Garvan reference).
%F a(n) = A066186(n) - A092269(n). - _Omar E. Pol_, Jan 08 2013
%F a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)) * (1 - (3*sqrt(6)/(2*Pi) + Pi/(24 * sqrt(6))) / sqrt(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-> 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_] := 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. Equals A220908/2.
%K nonn
%O 1,3
%A _N. J. A. Sloane_, Jan 02 2013
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