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A114725 The first entry of the vector v[n]=Mv[n-1], where M is the 6 X 6 matrix [[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1], [1, 5, 10, 10, 5, 1]] and v[0] is the column vector [0, 1, 1, 2, 3, 5].

%I A114725 #12 Jul 31 2015 18:13:56
%S A114725 0,1,1,2,3,5,55,136,502,1799,6247,21902,76882,269498,944895,3313259,
%T A114725 11617291,40733828,142826195,500794808,1755948163,6156922147,
%U A114725 21588159423,75695064671,265411365121,930618039799
%N A114725 The first entry of the vector v[n]=Mv[n-1], where M is the 6 X 6 matrix [[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1], [1, 5, 10, 10, 5, 1]] and v[0] is the column vector [0, 1, 1, 2, 3, 5].
%C A114725 Characteristic polynomial of the matrix M is x^6-(x+1)^5.
%H A114725 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1, 5, 10, 10, 5, 1).
%F A114725 Recurrence relation: a(n)=a(n-1)+5a(n-2)+10a(n-3)+10a(n-4)+5a(n-5)+a(n-6) for n>=6; a(0)=0,a(1)=a(2)=1,a(3)=2,a(4)=3,a(5)=5.
%F A114725 O.g.f.: x*(2*x+1)*(14*x^3+2*x-1)/(-1+x+5*x^2+10*x^3+10*x^4+5*x^5+x^6) . - _R. J. Mathar_, Dec 05 2007
%p A114725 a[0]:=0:a[1]:=1:a[2]:=1:a[3]:=2:a[4]:=3:a[5]:=5: for n from 6 to 25 do a[n]:=a[n-1]+5*a[n-2]+10*a[n-3]+10*a[n-4]+5*a[n-5]+a[n-6] od: seq(a[n],n=0..25);
%t A114725 M = {{0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0, 1}, {1, 4, 6, 4, 1}}; w[0] = {0, 1, 1, 2, 3}; w[n_] := w[n] = M.w[n - 1] a = Flatten[Table[w[n][[1]], {n, 0, 25}]]
%t A114725 LinearRecurrence[{1,5,10,10,5,1},{0,1,1,2,3,5},30] (* _Harvey P. Dale_, Jun 15 2014 *)
%K A114725 nonn
%O A114725 0,4
%A A114725 _Roger L. Bagula_, Feb 18 2006
%E A114725 Edited by _N. J. A. Sloane_, May 13 2006