A145840 Number of 4-compositions of n.

1, 4, 26, 164, 1031, 6480, 40728, 255984, 1608914, 10112368, 63558392, 399478064, 2510804924, 15780945024, 99186608832, 623409013632, 3918258753416, 24627092844352, 154786536605216, 972866430709568, 6114673231661936, 38432026791933696, 241553493927992448

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

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 (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

Cf. A003480, A145839, A145841.

Column k=4 of A261780.

Sequence in context: A121767 A092167 A124544 * A302335 A244787 A220305

Adjacent sequences: A145837 A145838 A145839 * A145841 A145842 A145843

Keyword

nonn,easy

Author

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

Extensions

Offset corrected by Alois P. Heinz, Aug 31 2015

Status

approved