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%I #46 Sep 08 2022 08:44:49
%S 1,2,4,7,14,25,50,91,182,336,672,1254,2508,4719,9438,17875,35750,
%T 68068,136136,260338,520676,999362,1998724,3848222,7696444,14858000,
%U 29716000,57500460,115000920,222981435,445962870,866262915,1732525830,3370764540
%N a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n and s(0) = 2. Also a(n) = sum of numbers in row n+1 of array T defined in A026009.
%C Conjecture: a(n) is the number of integer compositions of n + 2 in which the even parts appear as often at even positions as at odd positions (confirmed up to n = 19). - _Gus Wiseman_, Mar 17 2018
%H Michael De Vlieger, <a href="/A026010/b026010.txt">Table of n, a(n) for n = 0..3326</a>
%H Christian Krattenthaler, Daniel Yaqubi, <a href="https://arxiv.org/abs/1802.05990">Some determinants of path generating functions, II</a>, arXiv:1802.05990 [math.CO], 2018; Adv. Appl. Math. 101 (2018), 232-265.
%F a(2*n) = ((3*n + 1)/(2*n + 1))*C(2*n + 1, n)= A051924(1+n), n>=0, a(2*n-1) = a(2*n)/2 = A097613(1+n), n >= 1. - _Herbert Kociemba_, May 08 2004
%F a(n) = Sum_{k=0..n} binomial(floor((n+k)/2), floor(k/2)). - _Paul Barry_, Jul 15 2004
%F Inverse binomial transform of A005774: (1, 3, 9, 26, 75, 216, ...). - _Gary W. Adamson_, Oct 22 2007
%F Conjecture: (n+3)*a(n) - 2*a(n-1) + (-5*n-3)*a(n-2) + 2*a(n-3) + 4*(n-3)*a(n-4) = 0. - _R. J. Mathar_, Jun 20 2013
%F a(n) = (1/2)^((5 - (-1)^n)/2)*(6*n + 7 - 3*(-1)^n)*Catalan((2*n + 1 - (-1)^n)/4), where Catalan is the Catalan number = A000108. - _G. C. Greubel_, Nov 08 2018
%e The a(3) = 7 compositions of 5 in which the even parts appear as often at even positions as at odd positions are (5), (311), (131), (113), (221), (122), (11111). Missing are (41), (14), (32), (23), (212), (2111), (1211), (1121), (1112). - _Gus Wiseman_, Mar 17 2018
%t Array[Sum[Binomial[Floor[(# + k)/2], Floor[k/2]], {k, 0, #}] &, 34, 0] (* _Michael De Vlieger_, May 16 2018 *)
%t Table[2^(-1 + n)*(((2 + 3*#)*Gamma[(1 + #)/2])/(Sqrt[Pi]*Gamma[2 + #/2]) &[n + Mod[n, 2]]), {n,0,40}] (* Peter Pein, Nov 08 2018 *)
%t Table[(1/2)^((5 - (-1)^n)/2)*(6*n + 7 - 3*(-1)^n)*CatalanNumber[(2*n + 1 - (-1)^n)/4], {n, 0, 40}] (* _G. C. Greubel_, Nov 08 2018 *)
%o (PARI) vector(40, n, n--; sum(k=0,n, binomial(floor((n+k)/2), floor(k/2)))) \\ _G. C. Greubel_, Nov 08 2018
%o (Magma) [(&+[Binomial(Floor((n+k)/2), Floor(k/2)): k in [0..n]]): n in [0..40]]; // _G. C. Greubel_, Nov 08 2018
%Y First differences of A050168. Pairwise sums of A037952.
%Y Cf. A000712, A001405, A005774, A045931, A063886, A097613, A130780, A171966, A239241, A299926, A300061, A300787, A300788, A300789.
%K nonn
%O 0,2
%A _Clark Kimberling_
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