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A343386 Number of odd Motzkin n-paths, i.e., Motzkin n-paths with an odd number of up steps. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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: A018171 A306357 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

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Last modified December 9 05:44 EST 2021. Contains 349627 sequences. (Running on oeis4.)