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

0,4

Comments

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

Links

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.

Formula

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).

Lim_{n->infinity} 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

Example

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 + ...

Mathematica

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 *)

Prog

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

Crossrefs

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

Sequence in context: A163486 A214649 A007440 * A100223 A174017 A178081

Adjacent sequences: A343770 A343771 A343772 * A343774 A343775 A343776

Keyword

sign,easy

Author

Gennady Eremin, Sergey Kirgizov, Apr 29 2021

Status

approved