A211982 Second crank moment minus second rank moment: M_2(n) - N_2(n) = 2*spt(n).

2, 6, 10, 20, 28, 52, 70, 114, 160, 238, 322, 476, 630, 880, 1178, 1602, 2096, 2814, 3640, 4798, 6174, 7996, 10184, 13090, 16526, 20972, 26330, 33124, 41260, 51546, 63794, 79092, 97384, 119920, 146846, 179874, 219106, 266878, 323680, 392336, 473686

Offset: 1

Author

Omar E. Pol, Jan 03 2013

Keywords

• nonn

Comments

Also total number of smallest parts in all partitions of n, multiplied by 2.

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
• F. G. Garvan, Congruences for Andrews' smallest parts partition function and new congruences for Dyson's rank
• F. G. Garvan, Higher order spt-functions

Crossrefs

Cf. A092269, A066186, A220907, A220908, A220909.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1, n,
      `if`(irem(n, i, 'r')=0, r, 0)+add(b(n-i*j, i-1), j=0..n/i))
    end:
a:= n-> 2* b(n, n):
seq(a(n), n=1..60);  # Alois P. Heinz, Jan 17 2013

Mathematica

terms = 41; gf = Sum[x^n/(1 - x^n)*Product[1/(1 - x^k), {k, n, terms}], {n, 1, terms}]; 2*CoefficientList[ Series[gf, {x, 0, terms}], x] // Rest (* Jean-François Alcover, Jan 17 2013, from 2nd formula *)

Formula

a(n) = A220909(n) - A220908(n) = 2*A092269(n).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (Pi*sqrt(2*n)) * (1 - Pi/(24*sqrt(6*n)) + (144+Pi^2)/(6912*n)). - Vaclav Kotesovec, Jul 31 2017