A220908 The second rank moment function N_2(n).

0, 2, 8, 20, 42, 80, 140, 238, 380, 602, 910, 1372, 1996, 2900, 4102, 5790, 8002, 11046, 14980, 20282, 27090, 36092, 47546, 62510, 81374, 105700, 136210, 175084, 223510, 284694, 360410, 455244, 572054, 717160, 894964, 1114470, 1382032, 1710262, 2108750, 2594704, 3182120

Offset: 1

Author

N. J. A. Sloane, Jan 02 2013

Keywords

• nonn

Comments

N_2(n) is also called the second Atkin-Garvan moment (see Andrews' paper). - Omar E. Pol, Oct 23 2013

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• G. E. Andrews, The number of smallest parts in the partitions of n, p. 3
• F. G. Garvan, Higher-order spt functions, Adv. Math. 228 (2011), no. 1, 241-265. See Eq. (1.1).

Crossrefs

Cf. A000041, A092269, A220907.

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-> 2*(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_] := 2*(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) = 2*A220907(n) = 2*(n*A000041(n)-A092269(n)).

a(n) = 2*(A066186(n) - A092269(n)). - Omar E. Pol, Jan 09 2013

a(n) = A220909(n) - A211982(n). - Omar E. Pol, Jan 16 2013

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