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%I #29 Jun 29 2023 19:11:19
%S 1,2,5,17,65,257,1025,4097,16385,65537,262145,1048577,4194305,
%T 16777217,67108865,268435457,1073741825,4294967297,17179869185,
%U 68719476737,274877906945,1099511627777,4398046511105,17592186044417,70368744177665,281474976710657,1125899906842625,4503599627370497,18014398509481985
%N Row sums of A123162.
%H G. C. Greubel, <a href="/A123166/b123166.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5, -4).
%F a(n) = 1 + Sum_{k=0..n} binomial(2*n-1, 2*k-1), for n > 0. - _Paul Barry_, May 26 2008
%F a(n) = A052539(n-1), n > 0. - _R. J. Mathar_, Jun 18 2008
%F From _Sergei N. Gladkovskii_, Dec 20 2011: (Start)
%F G.f.: (1 - 3*x - x^2)/((1-x)*(1-4*x)).
%F E.g.f.: (exp(4*x) + 4*exp(x) - 1)/4 = (G(0) - 1)/4; G(k) = 1 + 4/(4^k-x*16^k/(x*4^k+(k+1)/G(k+1))); (continued fraction). (End)
%p a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=4*a[n-1] od: seq(a[n]+sum((k), k=0..1), n=0..20); # _Zerinvary Lajos_, Mar 20 2008
%t A123162[n_, k_]= If [k==0, 1, Binomial[2*n-1, 2*k-1]];
%t Table[Sum[A123162[n, k], {k,0,n}], {n,0,30}]
%t Table[4^(n-1) +1 -Boole[n==0]/4, {n,0,40}] (* _G. C. Greubel_, May 31 2022 *)
%o (Magma) [0] cat [4^(n-1) +1: n in [1..40]]; // _G. C. Greubel_, May 31 2022
%o (SageMath) [4^(n-1) +1 -bool(n==0)/4 for n in (0..40)] # _G. C. Greubel_, May 31 2022
%Y Cf. A052539, A123162.
%K nonn
%O 0,2
%A _Roger L. Bagula_, Oct 02 2006
%E Edited by _N. J. A. Sloane_, Oct 04 2006
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