cp's OEIS Frontend

A026122 a(n) is the number of (s(0),s(1),...,s(n)) such that every s(i) is a nonnegative integer, s(0) = 1, s(n) = 1, |s(1) - s(0)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = T(n,n), where T is the array in A026120.

%I A026122 #16 Jan 26 2022 08:54:16
%S A026122 2,4,11,28,74,196,525,1416,3846,10508,28864,79664,220818,614460,
%T A026122 1715874,4807008,13506534,38052972,107477319,304261404,863188662,
%U A026122 2453737132,6988033949,19935797080,56966012730,163026450132,467219178549,1340810339036
%N A026122 a(n) is the number of (s(0),s(1),...,s(n)) such that every s(i) is a nonnegative integer, s(0) = 1, s(n) = 1, |s(1) - s(0)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = T(n,n), where T is the array in A026120.
%F A026122 G.f.: (-1 + (1-z)^2 * M^2), with M the g.f. of the Motzkin numbers (A001006). [corrected by _Vaclav Kotesovec_, Sep 17 2019]
%F A026122 Conjecture: (n+4)*a(n) +(-3*n-5)*a(n-1) +(-n-6)*a(n-2) +3*(n-3)*a(n-3)=0. - _R. J. Mathar_, Jun 23 2013
%F A026122 a(n) ~ 4 * 3^(n + 1/2) / (sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Sep 17 2019
%F A026122 a(n) = 2*Sum_{m=1..n/2} C(2*m+1,m)*C(n-1,2*m-1)/(m+2). - _Vladimir Kruchinin_, Jan 24 2022
%t A026122 Drop[CoefficientList[Series[-1 + (1 - x)^2*(-1 + x + Sqrt[1 - 2*x - 3*x^2])^2 / (4*x^4), {x, 0, 30}], x], 2] (* _Vaclav Kotesovec_, Sep 17 2019 *)
%o A026122 (Maxima)
%o A026122 a(n):=2*sum((binomial(2*m+1,m)*binomial(n-1,2*m-1))/(m+2),m,1,n/2); /* _Vladimir Kruchinin_, Jan 24 2022 */
%Y A026122 Cf. A001006, A026120.
%Y A026122 First differences of A026107.
%K A026122 nonn
%O A026122 2,1
%A A026122 _Clark Kimberling_