A161434 Number of 6-compositions.

1, 6, 57, 524, 4803, 44022, 403495, 3698352, 33898338, 310705224, 2847860436, 26102905368, 239253883390, 2192952083712, 20100149570496, 184233853423936, 1688649759962676, 15477817777932456, 141866507103389516, 1300319342589168000, 11918460722228694720

Offset: 0

Author

Emanuele Munarini, Jun 10 2009

Keywords

• nonn
• easy

Comments

Excluding the terms of A161434(0) followed by the INVERTi transform yields A000389 without A000389(0). - Alexander R. Povolotsky and R. J. Mathar, Jun 16 2009

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• E. Grazzini, E. Munarini, M. Poneti, S. Rinaldi, m-compositions and m-partitions: exhaustive generation and Gray code, Pure Math. Appl. 17 (2006), 111-121.
• M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
• G. Louchard, Matrix Compositions: a Probabilistic analysis, Proc. GASCOM'08, Pure Mathematics and Applications, 19, 2-3, 127-146, 2008.
• E. Munarini, M. Poneti, S. Rinaldi, Matrix compositions, Journal of Integer Sequences, Vol. 12 (2009), Article 09.4.8
• Index entries for linear recurrences with constant coefficients, signature (12,-30,40,-30,12,-2).

Crossrefs

Column k=6 of A261780.

Cf. A000389, A161434.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-j)*binomial(j+5, 5), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 01 2015

Mathematica

Join[{1}, LinearRecurrence[{12, -30, 40, -30, 12, -2}, {6, 57, 524, 4803, 44022, 403495}, 20]] (* Jean-François Alcover, Jan 08 2016 *)
CoefficientList[Series[(1-x)^6/(2*(1-x)^6-1), {x, 0, 50}], x] (* G. C. Greubel, Nov 25 2017 *)

PARI

x='x+O('x^30); Vec((1-x)^6/(2*(1-x)^6-1)) \\ G. C. Greubel, Nov 25 2017

Formula

Recurrence: a(n+6) = 12*a(n+5) - 30*a(n+4) + 40*a(n+3) - 30*a(n+2) + 12*a(n+1) - 2*a(n).

G.f.: (1-x)^6/(2*(1-x)^6-1).

a(n) = Sum_{k>=0} (1/2)^(k+1) * binomial(n-1+6*k,n). - Seiichi Manyama, Aug 06 2024