This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A145840 #30 Nov 18 2024 16:14:56
%S A145840 1,4,26,164,1031,6480,40728,255984,1608914,10112368,63558392,
%T A145840 399478064,2510804924,15780945024,99186608832,623409013632,
%U A145840 3918258753416,24627092844352,154786536605216,972866430709568,6114673231661936,38432026791933696,241553493927992448
%N A145840 Number of 4-compositions of n.
%C A145840 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 A145840 G. Louchard, Matrix compositions: a probabilistic approach, Proceedings of GASCom and Bijective Combinatorics 2008, Bibbiena, Italy, pp. 159-170.
%D A145840 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 A145840 Alois P. Heinz, <a href="/A145840/b145840.txt">Table of n, a(n) for n = 0..1000</a>
%H A145840 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 A145840 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 A145840 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-12,8,-2).
%F A145840 a(n+4) = 8*a(n+3)-12*a(n+2)+8*a(n+1)-2*a(n).
%F A145840 G.f.: (1-x)^4/(2*(1-x)^4-1).
%F A145840 a(n) = sum(k>=0, C(n+4*k-1,n) / 2^(k+1)). - _Vaclav Kotesovec_, Dec 31 2013
%p A145840 a:= proc(n) option remember; `if`(n=0, 1,
%p A145840 add(a(n-j)*binomial(j+3, 3), j=1..n))
%p A145840 end:
%p A145840 seq(a(n), n=0..25); # _Alois P. Heinz_, Sep 01 2015
%t A145840 Table[Sum[Binomial[n+4*k-1,n]/2^(k+1),{k,0,Infinity}],{n,0,20}] (* _Vaclav Kotesovec_, Dec 31 2013 *)
%Y A145840 Cf. A003480, A145839, A145841.
%Y A145840 Column k=4 of A261780.
%K A145840 nonn,easy
%O A145840 0,2
%A A145840 Simone Rinaldi (rinaldi(AT)unisi.it), Oct 21 2008
%E A145840 Offset corrected by _Alois P. Heinz_, Aug 31 2015