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A116521 Binomial transform of tetranacci sequence A000078.

%I A116521 #16 Jun 28 2017 21:29:39
%S A116521 0,0,0,1,5,17,51,148,429,1250,3655,10701,31336,91752,268623,786414,
%T A116521 2302262,6739984,19731685,57765711,169112717,495088023,1449400960,
%U A116521 4243211207,12422263776,36366946961,106466490879,311687250156
%N A116521 Binomial transform of tetranacci sequence A000078.
%C A116521 See also A115390, the binomial transform of tribonacci sequence A000073. Tetranacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4), with a(0) = a(1) = a(2) = 0, a(3) = 1.
%H A116521 G. C. Greubel, <a href="/A116521/b116521.txt">Table of n, a(n) for n = 0..1000</a>
%H A116521 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,-8,6,-1).
%F A116521 a(n) = Sum_{k=0..n} C(n,k) * A000078(k).
%F A116521 G.f.: x^3/(1-5*x+8*x^2-6*x^3+x^4). - _Emeric Deutsch_, Apr 09 2006
%F A116521 a(n) = 5*a(n-1) - 8*a(n-2) + 6*a(n-3) - a(n-4). - _G. C. Greubel_, Nov 03 2016
%e A116521 Table shows the tetranacci numbers multiplied into rows of Pascal's triangle.
%e A116521 1*0 = 0.
%e A116521 1*0 + 1*0 = 0.
%e A116521 1*0 + 2*0 + 1*0 = 0.
%e A116521 1*0 + 3*0 + 3*0 + 1* 1 = 1.
%e A116521 1*0 + 4*0 + 6*0 + 4*1 + 1*1 = 5.
%e A116521 1*0 + 5*0 + 10*0 + 10*1 + 5*1 + 1*2 = 17.
%p A116521 t[0]:=0: t[1]:=0: t[2]:=0: t[3]:=1: for n from 4 to 35 do t[n]:=t[n-1]+t[n-2]+t[n-3]+t[n-4] od: seq(add(binomial(n,k)*t[k],k=0..n),n=0..30); # end of first Maple program
%p A116521 G:=x^3/(1-5*x+8*x^2-6*x^3+x^4): Gser:=series(G,x=0,33): seq(coeff(Gser,x,n),n=0..30); # _Emeric Deutsch_, Apr 09 2006
%t A116521 LinearRecurrence[{5,-8,6,-1}, {0,0,0,1}, 25] (* _G. C. Greubel_, Nov 03 2016 *)
%o A116521 (PARI) a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,6,-8,5]^n*[0;0;0;1])[1,1] \\ _Charles R Greathouse IV_, Jun 28 2017
%Y A116521 Cf. A000073, A000078, A115390.
%K A116521 easy,nonn
%O A116521 0,5
%A A116521 _Jonathan Vos Post_, Mar 10 2006
%E A116521 Definition corrected by _Franklin T. Adams-Watters_, Mar 13 2006
%E A116521 More terms from _Emeric Deutsch_, Apr 09 2006