A220907 a(n) = n*p(n)-spt(n) (= n*A000041(n) - A092269(n)).

0, 1, 4, 10, 21, 40, 70, 119, 190, 301, 455, 686, 998, 1450, 2051, 2895, 4001, 5523, 7490, 10141, 13545, 18046, 23773, 31255, 40687, 52850, 68105, 87542, 111755, 142347, 180205, 227622, 286027, 358580, 447482, 557235, 691016, 855131, 1054375, 1297352, 1591060

Offset: 1

Author

N. J. A. Sloane, Jan 02 2013

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• F. G. Garvan, Higher Order Spt-functions, Adv. Math. 228 (2011), no. 1, 241-265. See Eq. (1.1). - From _N. J. A. Sloane_, Jan 02 2013

Crossrefs

Cf. A000041, A092269. Equals A220908/2.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0 or i<1, 0,
      `if`(irem(n, i, 'r')=0, r, 0)+add(b(n-i*j, i-1), j=0..n/i))
    end:
a:= n-> n*combinat[numbpart](n)- b(n, n):
seq(a(n), n=1..60);  # Alois P. Heinz, Jan 09 2013

Mathematica

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) *)

Formula

a(n) = N_2(n)/2 (where N_2(n) is the second rank moment, see relation (1.1) of the Garvan reference).

a(n) = A066186(n) - A092269(n). - Omar E. Pol, Jan 08 2013

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