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

0,2

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

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, S. Rimaldi, Matrix compositions, JIS 12 (2009) 09.4.8

Index entries for linear recurrences with constant coefficients, signature (10,-20,20,-10,2).

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

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 *)

Crossrefs

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

Column k=5 of A261780.

Sequence in context: A144069 A280158 A073505 * A123943 A067412 A355355

Adjacent sequences: A145838 A145839 A145840 * A145842 A145843 A145844

Keyword

nonn,easy

Author

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

Extensions

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

Status

approved