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Binomial transform of (1,0,1,0,1,0,1,0,2,0,2,0,2,....).
1

%I #10 Mar 20 2023 06:54:33

%S 1,1,2,4,8,16,32,64,129,265,558,1200,2610,5682,12288,26292,55587,

%T 116179,240366,493108,1004780,2036692,4112144,8278552,16631717,

%U 33364381,66863358,133903816,268037862,536371734,1073120208,2146715436,4294024647,8588785575

%N Binomial transform of (1,0,1,0,1,0,1,0,2,0,2,0,2,....).

%C The sequence starting 1,2,4,... is the binomial transform of (1,1,1,1,1,1,1,2,2...) with a(n) = Sum_{k=0..6} C(n,k) + 2*Sum_{k=7..n} C(n,k) = 2^(n+1) - A008859(n). This gives the partial sums of A084637.

%H Colin Barker, <a href="/A084638/b084638.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (9,-35,77,-105,91,-49,15,-2).

%F a(n) = Sum_{k=0..3, C(n, 2*k)} + 2*Sum_{k=4..floor(n/2), C(n, 2*k)}.

%F a(n) = (n^6-15*n^5+115*n^4-405*n^3+964*n^2-660*n+720)/720 + 2*Sum_{k=4..floor(n/2), C(n, 2k)}.

%F G.f.: (1-8*x+28*x^2-56*x^3+70*x^4-56*x^5+28*x^6-8*x^7+2*x^8) / ((1-x)^7*(1-2*x)). - _Colin Barker_, Mar 17 2016

%t Table[2^n -4 -(1/6!)*(n+1)*(n^5-16*n^4+131*n^3-536*n^2+1500*n-2160) + Boole[n==0], {n,0,50}] (* _G. C. Greubel_, Mar 20 2023 *)

%o (PARI) Vec((1-8*x+28*x^2-56*x^3+70*x^4-56*x^5+28*x^6-8*x^7+2*x^8)/((1-x)^7*(1-2*x)) + O(x^50)) \\ _Colin Barker_, Mar 17 2016

%o (Magma) [2^n -4 -(n+1)*(n^5-16*n^4+131*n^3-536*n^2+1500*n-2160)/720 + 0^n: n in [0..50]]; // _G. C. Greubel_, Mar 20 2023

%o (SageMath) [2^n -4 -(n+1)*(n^5-16*n^4+131*n^3-536*n^2+1500*n-2160)/720 + 0^n for n in range(51)] # _G. C. Greubel_, Mar 20 2023

%Y Cf. A000225, A000325, A008859, A084634, A084635, A084636, A084637.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Jun 06 2003