%I #20 Mar 21 2023 09:21:29

%S 1,1,2,4,8,16,33,71,157,349,768,1662,3534,7398,15291,31297,63595,

%T 128555,258930,520240,1043540,2090956,4186757,8379499,16766313,

%U 33541481,67093588,134199826,268414602,536846754,1073713983,2147451717,4294930839,8589893143

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

%C Partial sums are A084637 (without leading 1).

%C The sequence starting 1,2,4,... is the binomial transform of (1,1,1,1,1,2,2,2,...) with b(n) = Sum_{k=0..4} C(n,k) + 2*Sum_{k=5..n} C(n,k) = 2^(n+1) - (n^4 -2*n^3 + 11*n^2 + 14*n + 24)/24. This gives the partial sums of A084635.

%H G. C. Greubel, <a href="/A084636/b084636.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (7,-20,30,-25,11,-2).

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

%F a(n) = (n^4 - 6*n^3 + 23*n^2 - 18*n + 24)/24 + 2*Sum_{k=3..floor(n/2)} C(n, 2*k).

%F O.g.f.: (1-2*x+2*x^2)*(1-4*x+5*x^2-2*x^3+x^4)/((1-x)^5*(1-2*x)). - _R. J. Mathar_, Apr 07 2008

%F a(n) = A000225(n) - (1/24)*n*(n-1)*(n^2 - 5*n + 18) + [n=0]. - _G. C. Greubel_, Mar 19 2023

%t Table[Boole[n==0] +(2^n-1) -(1/24)*n*(n^3-6*n^2+23*n-18), {n,0,50}] (* _G. C. Greubel_, Mar 19 2023 *)

%o (Magma) [(2^n-1) -(1/24)*n*(n^3-6*n^2+23*n-18) +0^n: n in [0..50]]; // _G. C. Greubel_, Mar 19 2023

%o (SageMath) [(2^n-1) -(1/24)*n*(n^3-6*n^2+23*n-18) +0^n for n in range(51)] # _G. C. Greubel_, Mar 19 2023

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

%K easy,nonn

%O 0,3

%A _Paul Barry_, Jun 06 2003