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

Author

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

Keywords

• nonn
• easy

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
• Sela Fried, Even-up words and their variants, arXiv:2505.14196 [math.CO], 2025. See p. 8.
• Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
• Emanuele Munarini, Maddalena Poneti, and Simone Rinaldi, Matrix compositions, JIS 12 (2009) 09.4.8.
• Index entries for linear recurrences with constant coefficients, signature (6,-6,2).

Crossrefs

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

Column k=3 of A261780.

Programs

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

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 *)

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

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

Extensions

Offset corrected by Alois P. Heinz, Aug 31 2015