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

Author

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

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A003480, A145839, A145841.

Column k=4 of A261780.

Programs

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 *)

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

Extensions

Offset corrected by Alois P. Heinz, Aug 31 2015