A145839 Number of 3-compositions of n.

1, 3, 15, 73, 354, 1716, 8318, 40320, 195444, 947380, 4592256, 22260144, 107902088, 523036176, 2535324816, 12289536016, 59571339552, 288761470848, 1399719859808, 6784893012864, 32888561860032, 159421452802624, 772767131681280, 3745851196992000

Offset

0,2

Comments

A 3-composition of n is a matrix with three rows, such that each column has at least one nonzero element and whose elements sum up to n.

Matrix inverse of (A000217(A004736)*A154990). - Mats Granvik, Jan 19 2009

(1 +3*x +15*x^2 +73*x^3 + ...) = 1/(1 -3*x -6*x^2 -10*x^3 -15*x^4 - ...). - Gary W. Adamson, Jul 27 2009

For n>1, a(n) is the number of generalized compositions of n-1 when there are i^2/2 +3i/2 +1 different types of i, (i=1,2,...). - Milan Janjic, Sep 24 2010

References

G. Louchard, Matrix compositions: a probabilistic approach, Proceedings of GASCom and Bijective Combinatorics 2008, Bibbiena, Italy, pp. 159-170.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.

E. Munarini, M. Poneti, and S. Rinaldi, Matrix compositions, JIS 12 (2009) 09.4.8.

Index entries for linear recurrences with constant coefficients, signature (6,-6,2).

Formula

a(n+3) = 6*a(n+2) - 6*a(n+1) + 2*a(n).

G.f.: (1-x)^3/(2*(1-x)^3 - 1).

a(n) = Sum_{k>=0} C(n+3*k-1,n) / 2^(k+1). - Vaclav Kotesovec, Dec 31 2013

a(n) = Sum_{j=0..n-1} binomial(n-j+2, 2)*a(j) with a(0) = 1. - G. C. Greubel, Mar 07 2021

Maple

a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-j)*binomial(j+2, 2), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 01 2015

Mathematica

Table[Sum[Binomial[n+3*k-1, n]/2^(k+1), {k, 0, Infinity}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 31 2013 *)
a[n_]:= a[n]= If[n==0, 1, Sum[Binomial[n-j+2, 2]*a[j], {j, 0, n-1}]]; Table[a[n], {n, 0, 20}] (* G. C. Greubel, Mar 07 2021 *)

Prog

(Sage)
@CachedFunction
def a(n):
    if n==0: return 1
    else: return sum( binomial(n-j+2, 2)*a(j) for j in (0..n-1))
[a(n) for n in (0..25)] # G. C. Greubel, Mar 07 2021
(Magma) I:=[3, 15, 73]; [1] cat [n le 3 select I[n] else 6*Self(n-1) - 6*Self(n-2) + 2*Self(n-3): n in [1..30]]; // G. C. Greubel, Mar 07 2021

Crossrefs

Cf. A003480 (2-compositions), A145840 (4-compositions), A145841 (5-compositions).

Column k=3 of A261780.

Sequence in context: A155117 A137638 A156019 * A232289 A370480 A055837

Adjacent sequences: A145836 A145837 A145838 * A145840 A145841 A145842

Keyword

nonn,easy

Author

Simone Rinaldi (rinaldi(AT)unisi.it), Oct 21 2008

Extensions

Offset corrected by Alois P. Heinz, Aug 31 2015

Status

approved