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

%I A211982 #31 Aug 16 2020 10:20:11
%S A211982 2,6,10,20,28,52,70,114,160,238,322,476,630,880,1178,1602,2096,2814,
%T A211982 3640,4798,6174,7996,10184,13090,16526,20972,26330,33124,41260,51546,
%U A211982 63794,79092,97384,119920,146846,179874,219106,266878,323680,392336,473686
%N A211982 Second crank moment minus second rank moment: M_2(n) - N_2(n) = 2*spt(n).
%C A211982 Also total number of smallest parts in all partitions of n, multiplied by 2.
%H A211982 Alois P. Heinz, <a href="/A211982/b211982.txt">Table of n, a(n) for n = 1..1000</a>
%H A211982 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>
%H A211982 F. G. Garvan, <a href="https://qseries.org/fgarvan/papers/spt.pdf">Congruences for Andrews' smallest parts partition function and new congruences for Dyson's rank</a>
%H A211982 F. G. Garvan, <a href="https://qseries.org/fgarvan/papers/hspt.pdf">Higher order spt-functions</a>
%F A211982 a(n) = A220909(n) - A220908(n) = 2*A092269(n).
%F A211982 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
%p A211982 b:= proc(n, i) option remember; `if`(n=0 or i=1, n,
%p A211982       `if`(irem(n, i, 'r')=0, r, 0)+add(b(n-i*j, i-1), j=0..n/i))
%p A211982     end:
%p A211982 a:= n-> 2* b(n, n):
%p A211982 seq(a(n), n=1..60);  # _Alois P. Heinz_, Jan 17 2013
%t A211982 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 *)
%Y A211982 Cf. A092269, A066186, A220907, A220908, A220909.
%K A211982 nonn
%O A211982 1,1
%A A211982 _Omar E. Pol_, Jan 03 2013