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 A145841 #30 Nov 18 2024 17:14:45
%S A145841 1,5,40,310,2395,18501,142920,1104060,8528890,65885880,508970002,
%T A145841 3931805460,30373291380,234634403620,1812556389540,14002041536004,
%U A145841 108166106338760,835585763004880,6454920038905520,49864411953151840,385203777033190008,2975708406629602400
%N A145841 Number of 5-compositions of n.
%C A145841 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.
%D A145841 G. Louchard, Matrix compositions: a probabilistic approach, Proceedings of GASCom and Bijective Combinatorics 2008, Bibbiena, Italy, pp. 159-170.
%D A145841 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 A145841 Alois P. Heinz, <a href="/A145841/b145841.txt">Table of n, a(n) for n = 0..1000</a>
%H A145841 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 A145841 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 A145841 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (10,-20,20,-10,2).
%F A145841 a(n+5) = 10*a(n+4)-20*a(n+3)+20*a(n+2)-10*a(n+1)+2*a(n).
%F A145841 G.f.: (1-x)^5/(2*(1-x)^5-1).
%F A145841 a(n) = sum(k>=0, C(n+5*k-1,n) / 2^(k+1)). - _Vaclav Kotesovec_, Dec 31 2013
%p A145841 a:= proc(n) option remember; `if`(n=0, 1,
%p A145841 add(a(n-j)*binomial(j+4, 4), j=1..n))
%p A145841 end:
%p A145841 seq(a(n), n=0..25); # _Alois P. Heinz_, Sep 01 2015
%t A145841 Table[Sum[Binomial[n+5*k-1,n]/2^(k+1),{k,0,Infinity}],{n,0,20}] (* _Vaclav Kotesovec_, Dec 31 2013 *)
%Y A145841 Cf. A003480 (2-compositions), A145839 (3-compositions), A145840 (4-compositions).
%Y A145841 Column k=5 of A261780.
%K A145841 nonn,easy
%O A145841 0,2
%A A145841 Simone Rinaldi (rinaldi(AT)unisi.it), Oct 21 2008
%E A145841 Offset changed from 1 to 0 by _Alois P. Heinz_, Aug 31 2015