%I #55 Sep 30 2021 18:11:55
%S 1,1,0,-2,-3,1,11,15,-13,-77,-86,144,595,495,-1520,-4810,-2485,15675,
%T 39560,6290,-159105,-324805,87075,1592843,2616757,-2136539,-15726114,
%U -20247800,32296693,152909577,145139491,-417959049,-1460704685,-885536173,4997618808,13658704994
%N Excess of the number of even Motzkin n-paths (A107587) over the odd ones (A343386).
%C All terms a(n), n >= 0, are contained in both A100223 and A214649, as well as in A007440 (if the signs of integers are not taken into account). So these sequences form a cluster, the base of which is the current sequence.
%C The Motzkin number A001006(n) is split into two parts A107587(n) and A343386(n) (see A343386). The value a(n), the difference between A107587(n) and A343386(n), can be called the "shadow" of A001006(n). This is clearly seen if we compare the g.f. for the Motzkin numbers M(x) = 1 + x*M(x) + x^2*M(x)^2 and the current g.f. A(x) = 1 + x*A(x) - x^2*A(x)^2.
%C Binomial transform of 1, 0, -1, 0, 2, 0, -5, 0, 14, 0, -42, 0, ... (see A000108). - _Gennady Eremin_, Jul 14 2021
%H Gennady Eremin, <a href="/A343773/b343773.txt">Table of n, a(n) for n = 0..800</a>
%H Gennady Eremin, <a href="https://arxiv.org/abs/2108.10676">Walking in the OEIS: From Motzkin numbers to Fibonacci numbers. The "shadows" of Motzkin numbers</a>, arXiv:2108.10676 [math.CO], 2021.
%F a(n) = A107587(n) - A343386(n), n>=0.
%F a(n) = A100223(n+2) = A214649(n+1), n>=0.
%F a(n) = (-1)^n * A007440(n+1), n>=0.
%F D-finite with recurrence a(n) = ((2*n+1)*a(n-1) - 5*(n-1)*a(n-2))/(n+2), n>1.
%F G.f.: (-1 + x + sqrt(1 - 2*x + 5*x^2))/(2*x^2).
%F G.f. A(x) satisfies A(x) = 1 + x*A(x) - x^2*A(x)^2.
%F a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(n, 2*k) * A000108(k).
%F a(n) = 2*A107587(n) - A001006(n) = A001006(n) - 2*A343386(n).
%F Lim_{n->infinity} a(n)/A001006(n) = 0.
%F a(n) = hypergeom([(1 - n)/2, -n/2], [2], -4). - _Peter Luschny_, May 30 2021
%F G.f. A(x) with offset 1 is the reversion of g.f. for signed Fibonacci numbers 1, -1, 2, -3, 5, -8, 13, -21, 34, -55, ... (see A039834 starting at offset 1). - _Gennady Eremin_, Jul 15 2021
%e G.f. = 1 + x - 2*x^3 - 3*x^4 + x^5 + 11*x^6 + 15*x^7 - 13*x^8 - 77*x^9 - 86*x^10 + 144*x^11 + ...
%t With[{$MaxExtraPrecision = 1000}, CoefficientList[Series[(-1 + x + Sqrt[1 - 2 x + 5 x^2])/(2 x^2), {x, 0, 36}], x] ] (* _Michael De Vlieger_, May 01 2021 *)
%t a[n_] := Hypergeometric2F1[(1 - n)/2, -n/2, 2, -4];
%t Table[a[n], {n, 0, 35}] (* _Peter Luschny_, May 30 2021 *)
%o (Python)
%o A343773 = [1, 1]
%o for n in range(2, 801):
%o A343773.append(((2*n+1)*A343773[-1]
%o - 5*(n-1)*A343773[-2]) // (n+2))
%Y Cf. A107587, A343386, A100223, A214649, A007440, A001006, A000108, A039834.
%K sign,easy
%O 0,4
%A _Gennady Eremin_, _Sergey Kirgizov_, Apr 29 2021