A085143 -id:A085143 - cp's OEIS frontend

A343773 Excess of the number of even Motzkin n-paths (A107587) over the odd ones (A343386).

1, 1, 0, -2, -3, 1, 11, 15, -13, -77, -86, 144, 595, 495, -1520, -4810, -2485, 15675, 39560, 6290, -159105, -324805, 87075, 1592843, 2616757, -2136539, -15726114, -20247800, 32296693, 152909577, 145139491, -417959049, -1460704685, -885536173, 4997618808, 13658704994

Offset: 0

Gennady Eremin, Sergey Kirgizov, Apr 29 2021

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.
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.
Binomial transform of 1, 0, -1, 0, 2, 0, -5, 0, 14, 0, -42, 0, ... (see A000108). - Gennady Eremin, Jul 14 2021
The Toeplitz Number Wall of a sequence a(0), a(1), ... is defined to be the 2D array A(m, n) of determinants of Toeplitz matrices a(i-j+n){i,j=1..m} where a(i) = 0 if i<0. Thus A(0, n) = 1, A(1, n) = a(n), A(2, n) = a(n)^2 - a(n+1)*a(n-1), A(m, n)^2 = A(m-1, n)*A(m+1, n) + A(m, n-1)*A(m, n+1) in general. For this sequence A(n, n) = F(n+1), A(n+1, n) = 1 and it is the unique sequence with this property. - _Michael Somos, Oct 27 2024

			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 + ...

Gennady Eremin, Table of n, a(n) for n = 0..800
Gennady Eremin, Walking in the OEIS: From Motzkin numbers to Fibonacci numbers. The "shadows" of Motzkin numbers, arXiv:2108.10676 [math.CO], 2021.
W. F. Lunnon, The number-wall algorithm: an LFSR cookbook, Journal of Integer Sequences 4 (2001), no. 1, 01.1.1.

Cf. A107587, A343386, A100223, A214649, A007440, A001006, A000108, A039834.

Cf. A085143.

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 *)
a[n_] := Hypergeometric2F1[(1 - n)/2, -n/2, 2, -4];
Table[a[n], {n, 0, 35}] (* Peter Luschny, May 30 2021 *)
a[ n_] := If[n<0, 0, SeriesCoefficient[Nest[1 + x*# - (x*#)^2&, 1 + O[x], n], {x, 0, n}]]; (* Michael Somos, Oct 27 2024 *)
a[ n_] := SeriesCoefficient[2/(1 - x + (1 - 2*x + 5*x^2)^(1/2)), {x, 0, n}]; (* Michael Somos, Oct 27 2024 *)

{a(n) = my(y = 1 + O(x)); for(i = 1, n, y = 1 + x*y - x^2*y^2); polcoeff(y, n)}; /* Michael Somos, Oct 27 2024 */

{a(n) = polcoeff( 2/(1 - x + (1 - 2*x + 5*x^2 + x*O(x^n))^(1/2)), n)}; /* Michael Somos, Oct 27 2024 */

A343773 = [1, 1]
for n in range(2, 801):
    A343773.append(((2*n+1)*A343773[-1]
      - 5*(n-1)*A343773[-2]) // (n+2))

a(n) = A107587(n) - A343386(n), n>=0.
a(n) = A100223(n+2) = A214649(n+1), n>=0.
a(n) = (-1)^n * A007440(n+1), n>=0.
D-finite with recurrence a(n) = ((2*n+1)*a(n-1) - 5*(n-1)*a(n-2))/(n+2), n>1.
G.f.: (-1 + x + sqrt(1 - 2*x + 5*x^2))/(2*x^2).
G.f. A(x) satisfies A(x) = 1 + x*A(x) - x^2*A(x)^2.
a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(n, 2*k) * A000108(k).
a(n) = 2*A107587(n) - A001006(n) = A001006(n) - 2*A343386(n).
Limit_{n->oo} a(n)/A001006(n) = 0.
a(n) = hypergeom([(1 - n)/2, -n/2], [2], -4). - Peter Luschny, May 30 2021
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

cp's OEIS Frontend

A343773 Excess of the number of even Motzkin n-paths (A107587) over the odd ones (A343386).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Formula