A145840 - OEIS

%I #30 Nov 18 2024 16:14:56

%S 1,4,26,164,1031,6480,40728,255984,1608914,10112368,63558392,

%T 399478064,2510804924,15780945024,99186608832,623409013632,

%U 3918258753416,24627092844352,154786536605216,972866430709568,6114673231661936,38432026791933696,241553493927992448

%N Number of 4-compositions of n.

%C 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.

%D G. Louchard, Matrix compositions: a probabilistic approach, Proceedings of GASCom and Bijective Combinatorics 2008, Bibbiena, Italy, pp. 159-170.

%D 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.

%H Alois P. Heinz, <a href="/A145840/b145840.txt">Table of n, a(n) for n = 0..1000</a>

%H Milan Janjić, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Janjic/janjic63.html">On Linear Recurrence Equations Arising from Compositions of Positive Integers</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.

%H E. Munarini, M. Poneti, and S. Rinaldi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Rinaldi/rinaldi.html">Matrix compositions</a>, JIS 12 (2009) 09.4.8.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-12,8,-2).

%F a(n+4) = 8*a(n+3)-12*a(n+2)+8*a(n+1)-2*a(n).

%F G.f.: (1-x)^4/(2*(1-x)^4-1).

%F a(n) = sum(k>=0, C(n+4*k-1,n) / 2^(k+1)). - _Vaclav Kotesovec_, Dec 31 2013

%p a:= proc(n) option remember; `if`(n=0, 1,

%p       add(a(n-j)*binomial(j+3, 3), j=1..n))

%p     end:

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, Sep 01 2015

%t Table[Sum[Binomial[n+4*k-1,n]/2^(k+1),{k,0,Infinity}],{n,0,20}] (* _Vaclav Kotesovec_, Dec 31 2013 *)

%Y Cf. A003480, A145839, A145841.

%Y Column k=4 of A261780.

%K nonn,easy

%O 0,2

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

%E Offset corrected by _Alois P. Heinz_, Aug 31 2015