A257178 -id:A257178 - cp's OEIS frontend

A257388 Number of 4-Motzkin paths of length n with no level steps at odd level.

1, 4, 17, 72, 306, 1304, 5573, 23888, 102702, 442904, 1915978, 8314480, 36195236, 158067312, 692475053, 3043191200, 13415404246, 59321085720, 263100680926, 1170347803440, 5221037429948, 23356788588752, 104772374565666, 471214329434208, 2124649562373708, 9603094073668208

Offset: 0

José Luis Ramírez Ramírez, Apr 21 2015

nonn

			For n=2 we have 17 paths: H(1)H(1), H(1)H(2), H(1)H(3), H(1)H(4), H(2)H(1), H(2)H(2), H(2)H(3), H(2)H(4), H(3)H(1), H(3)H(2), H(3)H(3), H(3)H(4), H(4)H(1), H(4)H(2), H(4)H(3), H(4)H(4) and UD.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A086622, A090344, A104498, A257178.

CoefficientList[Series[(1-4*x-Sqrt[(1-4*x)*(1-4*x-4*x^2)])/(2*x^2*(1-4*x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 22 2015 *)

x='x+O('x^50); Vec((1-4*x-sqrt((1-4*x)*(1-4*x-4*x^2)))/(2*x^2*(1-4*x))) \\ G. C. Greubel, Apr 08 2017

a(n) = Sum_{i=0..floor(n/2)}4^(n-2i)*C(i)*binomial(n-i,i), where C(n) is the n-th Catalan number A000108.
G.f.: (1-4*x-sqrt((1-4*x)*(1-4*x-4*x^2)))/(2*x^2*(1-4*x)).
a(n) ~ sqrt(58+41*sqrt(2)) * 2^(n+1/2) * (1+sqrt(2))^n / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 22 2015
Conjecture: (n+2)*a(n) +8*(-n-1)*a(n-1) +4*(3*n+1)*a(n-2) +8*(2*n-3)*a(n-3)=0. - R. J. Mathar, Sep 24 2016
G.f. A(x) satisfies: A(x) = 1/(1 - 4*x) + x^2 * A(x)^2. - Ilya Gutkovskiy, Jun 30 2020

A257389 Number of 3-generalized Motzkin paths of length n with no level steps H=(3,0) at odd level.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 6, 6, 17, 21, 54, 74, 183, 272, 644, 1025, 2342, 3928, 8734, 15264, 33227, 59989, 128484, 238008, 503563, 952038, 1995955, 3835381, 7987092, 15548654, 32223061, 63388488, 130918071, 259724317, 535168956, 1069025128

Offset: 0

José Luis Ramírez Ramírez, Apr 21 2015

nonn

			For n=6 we have 6 paths: UDUDUD, H3H3, UUDUDD, UUUDDD, UDUUDD and UUDDUD, where H3=(3,0).

Robert Israel, Table of n, a(n) for n = 0..3087

Cf. A000108, A090344, A086622, A257178, A007317, A064613, A104455, A104498.

f:= gfun:-rectoproc({(2 + n)*a(n) + (14 + 4*n)*a(n + 1) + (-10 - 2*n)*a(n + 3) + (-20 - 4*n)*a(n + 4) + (8 + n)*a(n + 6), a(0) = 1, a(1) = 0, a(2) = 1, a(3) = 1, a(4) = 2, a(5) = 2},a(n),remember):
map(f, [$0..100]); # Robert Israel, Nov 04 2019

a(n):=sum(((-1)^(n-3*k)+1)*((binomial((n-k)/2,k) )*(binomial(n-3*k,(n-3*k)/2))/((n-3*k+2))),k,0,(n)/3); /* Vladimir Kruchinin, Apr 02 2016 */

G.f.: (1-x^3-sqrt((1-x^3)*(1-4*x^2-x^3)))/(2*x^2*(1-x^3)).
a(n) = Sum_{k=0..n/3}(((-1)^(n-3*k)+1)*(binomial((n-k)/2,k)*(binomial(n-3*k,(n-3*k)/2))/((n-3*k+2)))). - Vladimir Kruchinin, Apr 02 2016
(2 + n)*a(n) + (14 + 4*n)*a(n + 1) + (-10 - 2*n)*a(n + 3) + (-20 - 4*n)*a(n + 4) + (8 + n)*a(n + 6) = 0. - Robert Israel, Nov 04 2019

A257515 Number of 3-generalized 2-Motzkin paths of length n with no level steps H=(3,0) at odd level.

Original entry on oeis.org

1, 0, 1, 2, 2, 4, 9, 12, 26, 48, 90, 172, 348, 664, 1349, 2680, 5438, 10976, 22510, 45900, 94700, 195032, 404442, 838824, 1748308, 3646368, 7632628, 15994232, 33606168, 70699504, 149050669, 314625264, 665280246, 1408436672, 2986069782, 6337988876

Offset: 0

José Luis Ramírez Ramírez, Apr 27 2015

nonn

			For n=6 we have 9 paths: UDUDUD, H3H3 (4 options), UUDUDD, UUUDDD, UDUUDD and UUDDUD, where H3=(3,0).

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A090344, A086622, A257178, A257388, A007317, A064613, A104455, A104498, A257389.

CoefficientList[Series[(1-2*x^3-Sqrt[(1-2x^3)*(1-4*x^2-2*x^3)])/(2*x^2*(1-2*x^3)), {x, 0, 30}], x] (* Vaclav Kotesovec, Apr 28 2015 *)

a(n):=sum((binomial(2*m,m)/(m+1)*(if mod(n+m,3)=0 then 2^((n-2*m)/3)* binomial((m+n)/3,m) else 0)),m,0,n); /* Vladimir Kruchinin, Mar 07 2016 */

seq(n)={Vec((1-2*x^3-sqrt((1-2*x^3)*(1-4*x^2-2*x^3) + O(x^(3+n))))/(2*x^2*(1-2*x^3)))} \\ Andrew Howroyd, May 01 2020

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

Conjecture: (n+2)*a(n) +(n+1)*a(n-1) +(n+4)*a(n-2) +4*(-2*n+3)*a(n-3) +4*(-6*n+17)*a(n-4) +4*(-3*n+10)*a(n-5) +4*(3*n-11)*a(n-6) +4*(11*n-50)*a(n-7) +20*(n-6)*a(n-8)=0. - R. J. Mathar, Jun 07 2016

Terms a(31) and beyond from Andrew Howroyd, May 01 2020

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A257388 Number of 4-Motzkin paths of length n with no level steps at odd level.

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A257389 Number of 3-generalized Motzkin paths of length n with no level steps H=(3,0) at odd level.

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A257515 Number of 3-generalized 2-Motzkin paths of length n with no level steps H=(3,0) at odd level.

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