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a(n) = p(n) - p(n-1) - p(n-2) + p(n-5), where p(n) = A000041(n).
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%I #36 Nov 05 2015 12:38:41

%S 1,0,0,0,0,0,0,-1,-1,-2,-3,-5,-6,-10,-13,-18,-24,-33,-42,-57,-72,-94,

%T -120,-154,-192,-245,-305,-382,-473,-588,-721,-891,-1087,-1330,-1617,

%U -1966,-2374,-2874,-3456,-4157,-4979,-5963,-7110,-8481,-10075,-11964,-14168

%N a(n) = p(n) - p(n-1) - p(n-2) + p(n-5), where p(n) = A000041(n).

%C In the introduction of the Andrews-Merca paper appears the inequality: p(n) - p(n-1) - p(n-2) + p(n-5) <= 0, for n > 0. Consider here that the partition number of a negative integer is equal to zero.

%C The absolute value of a(n) counts the partitions of n in which 2 is the least integer that is not a part and there are more parts >2 than there are 1. [_Mircea Merca_, Jul 13 2013]

%D M. Merca, Fast algorithm for generating ascending compositions, J. Math. Model. Algorithms 11 (2012), 89-104.

%H Alois P. Heinz, <a href="/A195054/b195054.txt">Table of n, a(n) for n = 0..10000</a>

%H G. E. Andrews and M. Merca, <a href="http://www.math.psu.edu/andrews/pdf/288.pdf">The Truncated Pentagonal Number Theorem</a>

%H G. E. Andrews and M. Merca, <a href="http://dx.doi.org/10.1016/j.jcta.2012.05.001">The truncated pentagonal number theorem</a>, J. Combin. Theory Ser. A, 119 (2012), 1639-1643.

%H M. Merca, <a href="http://dx.doi.org/10.1007/s10852-011-9168-y">Fast Algorithm for Generating Ascending Compositions</a> J. Math. Model. Algorithms, March 2012, Volume 11, Issue 1, pp 89-104 (DOI 10.1007/s10852-011-9168-y).

%F G.f.: (1 - x - x^2 + x^5) / Product_{k>=1} (1-x^k). - _Vaclav Kotesovec_, Nov 05 2015

%F a(n) ~ -Pi * exp(Pi*sqrt(2*n/3)) / (6 * sqrt(2) * n^(3/2)). - _Vaclav Kotesovec_, Nov 05 2015

%p p:= n-> `if`(n<0, 0, combinat[numbpart](n)):

%p a:= n-> p(n) -p(n-1) -p(n-2) +p(n-5):

%p seq(a(n), n=0..50); # _Alois P. Heinz_, Jan 22 2012

%t p = PartitionsP; a[n_] := p[n] - p[n-1] - p[n-2] + p[n-5]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Nov 05 2015 *)

%t nmax = 50; CoefficientList[Series[(1 - x - x^2 + x^5)/Product[1-x^k, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 05 2015 *)

%Y Cf. A000041, A002865.

%K sign,easy

%O 0,10

%A _Omar E. Pol_, Jan 14 2012

%E More terms from _Alois P. Heinz_, Jan 22 2012