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Expansion of (1+3*x)/(1-5*x).
4

%I #21 Mar 12 2024 02:47:47

%S 1,8,40,200,1000,5000,25000,125000,625000,3125000,15625000,78125000,

%T 390625000,1953125000,9765625000,48828125000,244140625000,

%U 1220703125000,6103515625000,30517578125000,152587890625000,762939453125000,3814697265625000,19073486328125000

%N Expansion of (1+3*x)/(1-5*x).

%C Binomial transform of A102900(n+1).

%C Hankel transform is := 1,-24,0,0,0,0,0,0,0,0,0,... - _Philippe Deléham_, Nov 02 2008

%H G. C. Greubel, <a href="/A128625/b128625.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (5).

%F a(n) = (8/5)*5^n - (3/5)*0^n.

%F a(0)=1, a(n) = Sum_{k=0..n} ((n+k)/n)*binomial(n,k)*2^(n-k)*3^k, n > 0.

%F E.g.f.: (8*exp(5*x) - 3)/5. - _G. C. Greubel_, Mar 12 2024

%t CoefficientList[Series[(1 + 3 x)/(1 - 5 x), {x, 0, 50}], x] (* _Vladimir Joseph Stephan Orlovsky_, Jun 21 2011 *)

%t Join[{1},NestList[5#&,8,30]] (* or *) LinearRecurrence[{5},{1,8},30] (* _Harvey P. Dale_, Dec 23 2021 *)

%o (Magma) [n eq 0 select 1 else 8*5^(n-1): n in [0..50]]; // _G. C. Greubel_, Mar 12 2024

%o (SageMath) [(8*5^n - 3*int(n==0))//5 for n in range(51)] # _G. C. Greubel_, Mar 12 2024

%Y Cf. A102900.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Mar 14 2007