|
%I #33 Sep 08 2022 08:44:49
%S 1,0,2,2,8,14,40,86,222,518,1296,3130,7770,19066,47324,117094,291260,
%T 724302,1806220,4507230,11266718,28188070,70609316,177023466,
%U 444231564,1115639586,2803975860,7052132546,17748069294,44693162266
%N a(n) = T(n,n), T given by A026584. Also a(n) is the number of integer strings s(0), ..., s(n) counted by T, such that s(n)=0.
%C The signed sequence 1,0,2,-2,8,-14,... is the inverse binomial transform of A026569. - _Paul Barry_, Sep 09 2004
%C Hankel transform of a(n) is 2^n. Hankel transform of a(n+1) is {0, -4, 0, 16, 0, -64, 0, 256, 0, ...} or -2^(n+1)*[x^n](x/(1+x^2)). Hankel transform of a(n+2) is 2^(n+1)*A109613(n+1). - _Paul Barry_, Mar 23 2011
%H Vincenzo Librandi, <a href="/A026585/b026585.txt">Table of n, a(n) for n = 0..1000</a>
%H J. W. Layman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL4/LAYMAN/hankel.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%F a(n) = A026584(n, n).
%F G.f.: sqrt((1-x)/(1-x-4*x^2)). - _Ralf Stephan_, Jan 08 2004
%F From _Paul Barry_, Jul 01 2009: (Start)
%F G.f.: 1/(1 -2*x^2/(1 -x -x^2/(1 -x^2/(1 -x -x^2/(1 -x^2/(1 -x -x^2/(1 - ... (continued fraction).
%F a(0) = 1, a(n) = Sum_{k=0..floor(n/2)} (k/(n-k))*C(n-k,k)*A000984(k). (End)
%F From _Paul Barry_, Mar 23 2011: (Start)
%F a(n) = Sum_{k=0..floor(n/2)} C(n-k-1,n-2*k)*A000984(k).
%F a(n) = Sum_{k=0..floor(n/2)} C(n-k-1,n-2*k)*C(2*k,k). (End)
%F D-finite with recurrence n*a(n) +2*(-n+1)*a(n-1) +(-3*n+2)*a(n-2) +2*(2*n-5)*a(n-3) = 0. - _R. J. Mathar_, Nov 24 2012
%F a(n) ~ (sqrt(17)+1)^(n-1/2) / (17^(1/4) * sqrt(Pi*n) * 2^(n-3/2)). - _Vaclav Kotesovec_, Feb 12 2014
%t CoefficientList[Series[Sqrt[(1-x)/(1-x-4*x^2)], {x, 0, 40}], x] (* _Vaclav Kotesovec_, Feb 12 2014 *)
%o (Magma) [(&+[Binomial(n-j-1, n-2*j)*Binomial(2*j, j): j in [0..Floor(n/2)]]): n in [0..40]]; // _G. C. Greubel_, Dec 12 2021
%o (Sage) [sum(binomial(n-j-1, n-2*j)*binomial(2*j, j) for j in (0..(n//2))) for n in [0..40]] # _G. C. Greubel_, Dec 12 2021
%Y Cf. A026584, A026587, A026589, A026590, A026591, A026592, A026593, A026594, A026595, A026596, A026597, A026598, A026599, A027282, A027283, A027284, A027285, A027286.
%Y Cf. A000984, A026569, A109613.
%K nonn
%O 0,3
%A _Clark Kimberling_
|
