A145841 Number of 5-compositions of n.

1, 5, 40, 310, 2395, 18501, 142920, 1104060, 8528890, 65885880, 508970002, 3931805460, 30373291380, 234634403620, 1812556389540, 14002041536004, 108166106338760, 835585763004880, 6454920038905520, 49864411953151840, 385203777033190008, 2975708406629602400

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

A 5-composition of n is a matrix with five rows, such that each column has at least one nonzero element and whose elements sum up to n.

References

• G. Louchard, Matrix compositions: a probabilistic approach, Proceedings of GASCom and Bijective Combinatorics 2008, Bibbiena, Italy, pp. 159-170.
• E. Munarini, M. Poneti and S. Rinaldi, Matrix compositions, Proceedings of Formal Power Series and Algebraic Combinatorics 2006, San Diego, USA, J. Remmel, M. Zabrocki (Editors) 445-456.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Milan Janjić, 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 (10,-20,20,-10,2).

Crossrefs

Cf. A003480 (2-compositions), A145839 (3-compositions), A145840 (4-compositions).

Column k=5 of A261780.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-j)*binomial(j+4, 4), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 01 2015

Mathematica

Table[Sum[Binomial[n+5*k-1,n]/2^(k+1),{k,0,Infinity}],{n,0,20}] (* Vaclav Kotesovec, Dec 31 2013 *)

Formula

a(n+5) = 10*a(n+4)-20*a(n+3)+20*a(n+2)-10*a(n+1)+2*a(n).

G.f.: (1-x)^5/(2*(1-x)^5-1).

a(n) = sum(k>=0, C(n+5*k-1,n) / 2^(k+1)). - Vaclav Kotesovec, Dec 31 2013

Extensions

Offset changed from 1 to 0 by Alois P. Heinz, Aug 31 2015