A195054 a(n) = p(n) - p(n-1) - p(n-2) + p(n-5), where p(n) = A000041(n).

1, 0, 0, 0, 0, 0, 0, -1, -1, -2, -3, -5, -6, -10, -13, -18, -24, -33, -42, -57, -72, -94, -120, -154, -192, -245, -305, -382, -473, -588, -721, -891, -1087, -1330, -1617, -1966, -2374, -2874, -3456, -4157, -4979, -5963, -7110, -8481, -10075, -11964, -14168

Offset

0,10

Comments

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.

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]

References

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

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

G. E. Andrews and M. Merca, The Truncated Pentagonal Number Theorem

G. E. Andrews and M. Merca, The truncated pentagonal number theorem, J. Combin. Theory Ser. A, 119 (2012), 1639-1643.

M. Merca, Fast Algorithm for Generating Ascending Compositions J. Math. Model. Algorithms, March 2012, Volume 11, Issue 1, pp 89-104 (DOI 10.1007/s10852-011-9168-y).

Formula

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

a(n) ~ -Pi * exp(Pi*sqrt(2*n/3)) / (6 * sqrt(2) * n^(3/2)). - Vaclav Kotesovec, Nov 05 2015

Maple

p:= n-> `if`(n<0, 0, combinat[numbpart](n)):
a:= n-> p(n) -p(n-1) -p(n-2) +p(n-5):
seq(a(n), n=0..50); # Alois P. Heinz, Jan 22 2012

Mathematica

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 *)
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 *)

Crossrefs

Cf. A000041, A002865.

Sequence in context: A018429 A341125 A035953 * A341126 A341127 A087750

Adjacent sequences: A195051 A195052 A195053 * A195055 A195056 A195057

Keyword

sign,easy

Author

Omar E. Pol, Jan 14 2012

Extensions

More terms from Alois P. Heinz, Jan 22 2012

Status

approved