A343386 Number of odd Motzkin n-paths, i.e., Motzkin n-paths with an odd number of up steps.

0, 0, 1, 3, 6, 10, 20, 56, 168, 456, 1137, 2827, 7458, 20670, 57577, 157691, 427976, 1170552, 3248411, 9096497, 25505562, 71436182, 200338074, 564083786, 1595055520, 4522769520, 12842772295, 36514010301, 103995490758, 296794937626, 848620165860, 2430089817720

Offset

0,4

Comments

a(n) is the number of Motzkin n-paths with an odd number of U-steps (see A001006). For example, there are 9 Motzkin 4-paths, of which six have one U-step each, namely: 00UD, 0U0D, 0UD0, U00D, U0D0, and UD00. So a(4) = 6.

Number of Motzkin n-paths that, after removing the horizontal steps, are converted to Dyck (2m)-paths, where 2m <= n and m is odd (see A024492).

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) = Sum_{k=0..n} binomial(n, 4*k+2) * A000108(2*k+1).

a(n) = A001006(n) - A107587(n).

G.f.: A(x) = (2 - 2*x - sqrt(1-2*x-3*x^2) - sqrt(1-2*x+5*x^2))/(4*x^2).

G.f. A(x) satisfies A(x) = x*A(x) + x^2*A(x)^2 + x^2*B(x)^2 where B(x) is the g.f. of A107587.

a(n) = A107587(n) - A100223(n+2). - R. J. Mathar, Apr 16 2021

D-finite with recurrence: n*(n+2)*a(n) + (-5*n^2-n+3)*a(n-1) + (10*n^2-16*n+3)*a(n-2) + (-10*n^2+34*n-27)*a(n-3) - (11*n-5)*(n-3)*a(n-4) + 15*(n-3)*(n-4)*a(n-5) = 0, n >= 5. - R. J. Mathar, Apr 17 2021

D-finite with recurrence: n*(n-2)*(n+2)*a(n) - (2*n-1)*(2*n^2-2*n-3)*a(n-1) + 3*(n-1)*(2*n^2-4*n+1)*a(n-2) - 2*(n-1)*(n-2)*(2*n-3)*a(n-3) - 15*(n-1)*(n-2)*(n-3)*a(n-4) = 0, n >= 4. - R. J. Mathar, Apr 17 2021

a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k) * A000108(k) * (k mod 2). - Gennady Eremin, May 03 2021 [after Paul Barry (A107587)]

a(n) = ((n-1)*n*hypergeom([1/2-n/4, 3/4-n/4, 1-n/4, 5/4-n/4], [3/2, 3/2, 2], 16))/2. - Peter Luschny, Sep 24 2021

a(n) ~ 3^(n + 3/2) / (4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 27 2024

Example

G.f. = x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 20*x^6 + 56*x^7 + 168*x^8 + ...

Mathematica

a[n_] := ((n - 1) n HypergeometricPFQ[{1/2 - n/4, 3/4 - n/4, 1 - n/4, 5/4 - n/4}, {3/2, 3/2, 2}, 16])/2;
Table[a[n], {n, 0, 31}] (* Peter Luschny, Sep 24 2021 *)

Prog

(Python)
M = [4, 9]; E = [1, 1, 1, 1, 3];
A343386 = [0, 0, 1, 3, 6]
for n in range(5, 801):
    M.append(((2*n+1)*M[1]+(3*n-3)*M[0])//(n+2))
    E.append(((5*n**2+n-3)*E[4] - (10*n**2-16*n+3)*E[3]
      + (10*n**2-34*n+27)*E[2] + (11*n-5)*(n-3)*E[1]
      - 15*(n-3)*(n-4)*E[0]) // (n*n+2*n))
    A343386.append(M[-1] - E[-1])
    M.pop(0); E.pop(0)

Crossrefs

Cf. A000108, A001006, A107587, A024492, A100223.

Sequence in context: A306357 A365694 A122628 * A068865 A060179 A056411

Adjacent sequences: A343383 A343384 A343385 * A343387 A343388 A343389

Keyword

nonn,easy

Author

Gennady Eremin, Sergey Kirgizov, Apr 13 2021

Status

approved