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%I #12 Mar 10 2019 17:01:57
%S 5,6,10,24,70,216,664,2008,5998,17808,52770,156360,463492,1374392,
%T 4076222,12090144,35859742,106359928,315460168,935639768,2775057510,
%U 8230670416,24411730298,72403913480,214746249796,636926269816
%N Binomial transform of pentanacci numbers A074048: a(n) = Sum_{k=0..n} binomial(n,k)*A074048(k).
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (6,-13,14,-7,2).
%F a(n) = 6a(n-1) - 13a(n-2) + 14a(n-3) - 7a(n-4) + 2a(n-5), a(0)=5, a(1)=6, a(2)=10, a(3)=24, a(4)=70.
%F G.f.: (5 - 24*x + 39*x^2 - 28*x^3 + 7*x^4)/(1 - 6*x + 13*x^2 - 14*x^3 + 7*x^4 - 2*x^5).
%F a(n) = term (1,5) in the 1 X 5 matrix [70,24,10,6,5]. [6,1,0,0,0; -13,0,1,0,0; 14,0,0,1,0; -7,0,0,0,1; 2,0,0,0,0]^n. - _Alois P. Heinz_, Jul 25 2008
%p M := Matrix(5, (i,j)-> if (i=j-1) then 1 elif j>1 then 0 else [6,-13,14,-7,2][i] fi); a := n -> (Matrix([[70,24,10,6,5]]).M^(n))[1,5]; seq (a(n), n=0..50); # _Alois P. Heinz_, Jul 25 2008
%t CoefficientList[Series[(5-24*x+39*x^2-28*x^3+7*x^4)/(1-6*x+13*x^2-14*x^3+7*x^4-2*x^5), {x, 0, 25}], x]
%t LinearRecurrence[{6,-13,14,-7,2},{5,6,10,24,70},30] (* _Harvey P. Dale_, Mar 10 2019 *)
%Y Cf. A074048.
%K easy,nonn
%O 0,1
%A Mario Catalani (mario.catalani(AT)unito.it), Sep 07 2002
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