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A145841
Number of 5-compositions of n.
6
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
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
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
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