cp's OEIS Frontend

A026010 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.

%I A026010 #46 Sep 08 2022 08:44:49
%S A026010 1,2,4,7,14,25,50,91,182,336,672,1254,2508,4719,9438,17875,35750,
%T A026010 68068,136136,260338,520676,999362,1998724,3848222,7696444,14858000,
%U A026010 29716000,57500460,115000920,222981435,445962870,866262915,1732525830,3370764540
%N A026010 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 A026010 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 A026010 Michael De Vlieger, <a href="/A026010/b026010.txt">Table of n, a(n) for n = 0..3326</a>
%H A026010 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 A026010 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 A026010 a(n) = Sum_{k=0..n} binomial(floor((n+k)/2), floor(k/2)). - _Paul Barry_, Jul 15 2004
%F A026010 Inverse binomial transform of A005774: (1, 3, 9, 26, 75, 216, ...). - _Gary W. Adamson_, Oct 22 2007
%F A026010 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 A026010 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 A026010 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 A026010 Array[Sum[Binomial[Floor[(# + k)/2], Floor[k/2]], {k, 0, #}] &, 34, 0] (* _Michael De Vlieger_, May 16 2018 *)
%t A026010 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 A026010 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 A026010 (PARI) vector(40, n, n--; sum(k=0,n, binomial(floor((n+k)/2), floor(k/2)))) \\ _G. C. Greubel_, Nov 08 2018
%o A026010 (Magma) [(&+[Binomial(Floor((n+k)/2), Floor(k/2)): k in [0..n]]): n in [0..40]]; // _G. C. Greubel_, Nov 08 2018
%Y A026010 First differences of A050168. Pairwise sums of A037952.
%Y A026010 Cf. A000712, A001405, A005774, A045931, A063886, A097613, A130780, A171966, A239241, A299926, A300061, A300787, A300788, A300789.
%K A026010 nonn
%O A026010 0,2
%A A026010 _Clark Kimberling_