The On-Line Encyclopedia of Integer Sequences (OEIS)

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A257290

Number of 3-Motzkin paths of length n with no level steps at even level.
(history; published version)

#27 by Charles R Greathouse IV at Tue Feb 14 22:11:30 EST 2017

STATUS

proposed

approved

#26 by G. C. Greubel at Tue Feb 14 17:53:08 EST 2017

STATUS

editing

proposed

#25 by G. C. Greubel at Tue Feb 14 17:52:56 EST 2017

DATA

1, 0, 1, 3, 11, 39, 140, 504, 1823, 6621, 24144, 88380, 324699, 1197045, 4427565, 16427385, 61129025, 228103185, 853399640, 3200710680, 12032399045, 45332769075, 171148151095, 647412581643, 2453529142471, 9314461044639, 35419207688050, 134894888442714, 514506926871927

LINKS

G. C. Greubel, <a href="/A257290/b257290.txt">Table of n, a(n) for n = 0..1000</a>

FORMULA

a(n) = Sum_{i=0..floor(n/2)} 3^(n-2i)*C(i)*binomial(n-i-1,n), where C(i) is the n-th Catalan number A000108.

G.f.: (1 -3z 3*z - sqrt((1-3z3*z)*(1-3z3*z-4z4*z^2)))/(2z2*z^2).

PROG

(PARI) x='x+O('x^50); Vec((1-3*x-sqrt((1-3*x)*(1-3*x-4*x^2)))/(2*x^2)) \\ G. C. Greubel, Feb 14 2017

STATUS

approved

editing

#24 by R. J. Mathar at Sat Sep 24 18:19:57 EDT 2016

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editing

approved

#23 by R. J. Mathar at Sat Sep 24 18:19:52 EDT 2016

FORMULA

Conjecture: (n+2)*a(n) +3*(-2*n-1)*a(n-1) +5*(n-1)*a(n-2) +6*(2*n-5)*a(n-3)=0. - R. J. Mathar, Sep 24 2016

STATUS

approved

editing

#22 by Vaclav Kotesovec at Tue Apr 21 04:45:19 EDT 2015

STATUS

editing

approved

#21 by Vaclav Kotesovec at Tue Apr 21 04:45:12 EDT 2015

FORMULA

a(n) = Sum_{i=0..floor(n/2)}3^(n-2i)*C(i)*binomial(n-i-1,n), where C(i) is the n-th Catalan number A000108.

a(n) ~ sqrt(5) * 4^n / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 21 2015

#20 by Vaclav Kotesovec at Tue Apr 21 04:39:39 EDT 2015

FORMULA

G.f.: (1-3z-sqrt((1-3z)(1-3z-4z^2)))/(2z^2).

MATHEMATICA

CoefficientList[Series[(1-3*x-Sqrt[(1-3*x)*(1-3*x-4*x^2)])/(2*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 21 2015 *)

STATUS

approved

editing

#19 by N. J. A. Sloane at Tue Apr 21 01:07:14 EDT 2015

STATUS

proposed

approved

#18 by José Luis Ramírez Ramírez at Mon Apr 20 22:47:01 EDT 2015

STATUS

editing

proposed