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 A161434 #33 Aug 06 2024 09:56:45
%S A161434 1,6,57,524,4803,44022,403495,3698352,33898338,310705224,2847860436,
%T A161434 26102905368,239253883390,2192952083712,20100149570496,
%U A161434 184233853423936,1688649759962676,15477817777932456,141866507103389516,1300319342589168000,11918460722228694720
%N A161434 Number of 6-compositions.
%C A161434 Excluding the terms of A161434(0) followed by the INVERTi transform yields A000389 without A000389(0). - _Alexander R. Povolotsky_ and _R. J. Mathar_, Jun 16 2009
%H A161434 Alois P. Heinz, <a href="/A161434/b161434.txt">Table of n, a(n) for n = 0..1000</a>
%H A161434 E. Grazzini, E. Munarini, M. Poneti, S. Rinaldi, <a href="http://www.mat.unisi.it/newsito/puma/public_html/17_1_2/grazzini.pdf">m-compositions and m-partitions: exhaustive generation and Gray code</a>, Pure Math. Appl. 17 (2006), 111-121.
%H A161434 M. Janjic, <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 A161434 G. Louchard, <a href="http://www.ulb.ac.be/di/mcs/louchard/louchard.papers/compmat.ps">Matrix Compositions: a Probabilistic analysis</a>, Proc. GASCOM'08, Pure Mathematics and Applications, 19, 2-3, 127-146, 2008.
%H A161434 E. Munarini, M. Poneti, S. Rinaldi, <a href="http://www.emis.de/journals/JIS/VOL12/Rinaldi/rinaldi.html">Matrix compositions</a>, Journal of Integer Sequences, Vol. 12 (2009), Article 09.4.8
%H A161434 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (12,-30,40,-30,12,-2).
%F A161434 Recurrence: a(n+6) = 12*a(n+5) - 30*a(n+4) + 40*a(n+3) - 30*a(n+2) + 12*a(n+1) - 2*a(n).
%F A161434 G.f.: (1-x)^6/(2*(1-x)^6-1).
%F A161434 a(n) = Sum_{k>=0} (1/2)^(k+1) * binomial(n-1+6*k,n). - _Seiichi Manyama_, Aug 06 2024
%p A161434 a:= proc(n) option remember; `if`(n=0, 1,
%p A161434 add(a(n-j)*binomial(j+5, 5), j=1..n))
%p A161434 end:
%p A161434 seq(a(n), n=0..25); # _Alois P. Heinz_, Sep 01 2015
%t A161434 Join[{1}, LinearRecurrence[{12, -30, 40, -30, 12, -2}, {6, 57, 524, 4803, 44022, 403495}, 20]] (* _Jean-François Alcover_, Jan 08 2016 *)
%t A161434 CoefficientList[Series[(1-x)^6/(2*(1-x)^6-1), {x, 0, 50}], x] (* _G. C. Greubel_, Nov 25 2017 *)
%o A161434 (PARI) x='x+O('x^30); Vec((1-x)^6/(2*(1-x)^6-1)) \\ _G. C. Greubel_, Nov 25 2017
%Y A161434 Column k=6 of A261780.
%Y A161434 Cf. A000389, A161434.
%K A161434 nonn,easy
%O A161434 0,2
%A A161434 _Emanuele Munarini_, Jun 10 2009