OFFSET
0,2
COMMENTS
A 4-composition of n is a matrix with four 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 (8,-12,8,-2).
FORMULA
a(n+4) = 8*a(n+3)-12*a(n+2)+8*a(n+1)-2*a(n).
G.f.: (1-x)^4/(2*(1-x)^4-1).
a(n) = sum(k>=0, C(n+4*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+3, 3), j=1..n))
end:
seq(a(n), n=0..25); # Alois P. Heinz, Sep 01 2015
MATHEMATICA
Table[Sum[Binomial[n+4*k-1, n]/2^(k+1), {k, 0, Infinity}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 31 2013 *)
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
Simone Rinaldi (rinaldi(AT)unisi.it), Oct 21 2008
EXTENSIONS
Offset corrected by Alois P. Heinz, Aug 31 2015
STATUS
approved