José Luis Ramírez Ramírez - cp's OEIS frontend

A322378 Triangle read by rows: T(n,k) is the number of nondecreasing Dyck prefixes (i.e., left factors of nondecreasing Dyck paths) of length n and final height k (0 <= k <= n).

1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 2, 0, 3, 0, 1, 0, 5, 0, 4, 0, 1, 5, 0, 9, 0, 5, 0, 1, 0, 13, 0, 14, 0, 6, 0, 1, 13, 0, 26, 0, 20, 0, 7, 0, 1, 0, 34, 0, 45, 0, 27, 0, 8, 0, 1, 34, 0, 73, 0, 71, 0, 35, 0, 9, 0, 1, 0, 89, 0, 137, 0, 105, 0, 44, 0, 10, 0, 1, 89, 0, 201, 0, 234, 0, 148, 0, 54, 0, 11, 0, 1, 0, 233, 0, 402, 0, 373, 0, 201, 0, 65, 0, 12, 0, 1, 233, 0, 546, 0, 733, 0, 564, 0, 265, 0, 77, 0, 13, 0, 1, 0, 610, 0, 1149, 0, 1245, 0, 818, 0, 341, 0, 90, 0, 14, 0, 1

Offset: 0

José Luis Ramírez Ramírez, Dec 05 2018

			Triangle begins:
   1;
   0,   1;
   1,   0,   1;
   0,   2,   0,   1;
   2,   0,   3,   0,   1;
   0,   5,   0,   4,   0,   1;
   5,   0,   9,   0,   5,   0,   1;
   0,  13,   0,  14,   0,   6,   0,   1;
  13,   0,  26,   0,  20,   0,   7,   0,   1;
   0,  34,   0,  45,   0,  27,   0,   8,   0,   1;
  34,   0,  73,   0,  71,   0,  35,   0,   9,   0,   1;
   0,  89,   0, 137,   0, 105,   0,  44,   0,  10,   0,   1;
  89,   0, 201,   0, 234,   0, 148,   0,  54,   0,  11,   0,   1;
   0, 233,   0, 402,   0, 373,   0, 201,   0,  65,   0,  12,   0,   1;
  ...

R. Flórez and J. L. Ramírez, Some enumerations on non-decreasing Motzkin paths, Australasian Journal of Combinatorics, 72(1) (2018), 138-154.

Columns k=0, 1 give A001519. Column k=2 gives A061667.

Cf. A322329, A322325.

Riordan array: ((1 - 2*x^2)/(1 - 3*x^2 + x^4), (x*(1-x^2))/(1 - 2*x^2)).

A322329 Triangle read by rows: T(n,k) is the number of nondecreasing Motzkin prefixes (i.e., left factors of nondecreasing Motzkin paths) of length n and final height k (0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 5, 3, 1, 9, 12, 9, 4, 1, 21, 30, 25, 14, 5, 1, 49, 74, 69, 44, 20, 6, 1, 115, 182, 185, 133, 70, 27, 7, 1, 269, 444, 488, 386, 230, 104, 35, 8, 1, 630, 1078, 1266, 1090, 718, 369, 147, 44, 9, 1, 1474, 2605, 3245, 3006, 2161, 1232, 560, 200, 54, 10, 1

Offset: 0

José Luis Ramírez Ramírez, Dec 03 2018

			Triangle begins:
     1;
     1,    1;
     2,    2,    1;
     4,    5,    3,    1;
     9,   12,    9,    4,    1;
    21,   30,   25,   14,    5,    1;
    49,   74,   69,   44,   20,    6,   1;
   115,  182,  185,  133,   70,   27,   7,   1;
   269,  444,  488,  386,  230,  104,  35,   8,  1;
   630, 1078, 1266, 1090,  718,  369, 147,  44,  9,  1;
  1474, 2605, 3245, 3006, 2161, 1232, 560, 200, 54, 10, 1;
  ...

R. Flórez and J. L. Ramírez, Some enumerations on non-decreasing Motzkin paths, Australasian Journal of Combinatorics, 72(1) (2018), 138-154.

Column k=0 gives A322325.

Cf. A097862, A283595.

Riordan array: ((1 - x - 2*x^2 + x^3)/(1 - 2*x - 2*x^2 + 3x^3 - x^5),(x*(1-x)^2*(1+x))/(1 - 2*x - x^2 + 2*x^3 - x^4)).

A322325 Number of nondecreasing Motzkin paths of length n.

Original entry on oeis.org

1, 1, 2, 4, 9, 21, 49, 115, 269, 630, 1474, 3450, 8073, 18893, 44212, 103465, 242125, 566617, 1325982, 3103035, 7261648, 16993545, 39767898, 93063924, 217786044, 509657890, 1192689641, 2791104828, 6531679192, 15285285161, 35770272112, 83708766611, 195893326791

Offset: 0

José Luis Ramírez Ramírez, Dec 03 2018

			For n=6 we have 49 paths. Among the A001006(6) = 51 Motzkin paths, the following two paths are not nondecreasing Motzkin paths: UHUDDH and UUDHDH.

R. Flórez and J. L. Ramírez, Some enumerations on non-decreasing Motzkin paths, Australasian Journal of Combinatorics, 72(1) (2018), 138-154.
Index entries for linear recurrences with constant coefficients, signature (2,2,-3,0,1)

Column k=0 of A322329.

Cf. A001006, A001519.

LinearRecurrence[{2, 2, -3, 0, 1}, {1, 1, 2, 4, 9}, 40] (* Amiram Eldar, Dec 03 2018 *)

a(n) = 2*a(n-1) + 2*a(n-2) - 3*a(n-3) + a(n-5), a(0)=1, a(1)=1, a(2)=2, a(3)=4, a(4)=9.

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

A253831 Number of 2-Motzkin paths with no level steps at height 1.

Original entry on oeis.org

1, 2, 5, 12, 30, 76, 197, 522, 1418, 3956, 11354, 33554, 102104, 319608, 1027237, 3381714, 11371366, 38946892, 135505958, 477781296, 1703671604, 6132978608, 22256615602, 81327116484, 298938112816, 1104473254912, 4098996843500, 15272792557230, 57106723430892, 214202598271360, 805743355591301

Offset: 0

José Luis Ramírez Ramírez, Apr 20 2015

nonn

For n=3 we have 12 paths: H(1)H(1)H(1), H(1)H(1)H(2), H(1)H(2)H(1), H(1)H(2)H(2), H(2)H(1)H(1), H(2)H(1)H(2), H(2)H(2)H(1), H(2)H(2)H(2), UDH(1), UDH(2), H(1)UD, H(2)UD.

Robert Israel, Table of n, a(n) for n = 0..1000

Cf. A000957, A217312, A257363.

rec:= (54+36*n)*a(n)+(-3+7*n)*a(n+1)+(-60-36*n)*a(n+2)+(36+16*n)*a(n+3)+(-6-2*n)*a(n+4) = 0:
f:= gfun:-rectoproc({rec,seq(a(i)=[1,2,5,12][i+1],i=0..3)},a(n),remember):
seq(f(n),n=0..100); # Robert Israel, Apr 29 2015

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

a(n):=sum(sum(((sum((k+1)*binomial(k+m,k+1)*binomial(2*j-k+m-1,j-k)*(-1)^(k),k,0,j))*2^(n-j-2*m)*binomial(n-m-j,m))/(j+m),j,0,n-2*m),m,1,n/2)+2^n; /* Vladimir Kruchinin, Mar 11 2016 */

G.f.: 1/(1-2*x-x*F(x)), where F(x) is the g.f. of Fine numbers A000957.
G.f.: 2*(2+x)/(4-7*x-6*x^2+x*sqrt(1-4*x)).
a(n) ~ 4^(n+1) / (25*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 21 2015
(54+36*n)*a(n)+(-3+7*n)*a(n+1)+(-60-36*n)*a(n+2)+(36+16*n)*a(n+3)+(-6-2*n)*a(n+4) = 0. - Robert Israel, Apr 29 2015
a(n) = Sum_{m=1..n/2}(Sum_{j=0..n-2*m}(((Sum_{k=0..j}((k+1)*binomial(k+m,k+1)*binomial(2*j-k+m-1,j-k)*(-1)^(k)))*2^(n-j-2*m)*binomial(n-m-j,m))/(j+m)))+2^n. - Vladimir Kruchinin, Mar 11 2016

A257390 Number of 4-Motzkin paths of length n with no level steps at even level.

Original entry on oeis.org

1, 0, 1, 4, 18, 80, 357, 1596, 7150, 32096, 144362, 650568, 2937316, 13286368, 60205805, 273290988, 1242639446, 5659468736, 25816338046, 117945079736, 539646216188, 2472638868960, 11345220210658, 52124831171544, 239792244636876, 1104495824173376

Offset: 0

José Luis Ramírez Ramírez, Apr 21 2015

nonn

Robert Israel, Table of n, a(n) for n = 0..1322

Cf. A090345, A025266, A257290.

rec:= a(n) = ((-16*n + 40)*a(n-3) + (-12*n+12)*a(n-2) +(8*n+4)*a(n-1))/(n+2):
f:= gfun:-rectoproc({rec,a(0)=1,a(1)=0,a(2)=1},a(n),remember):
seq(f(i),i=0..100); # Robert Israel, Apr 22 2015

CoefficientList[Series[(1-4*x-Sqrt[(1-4*x)*(1-4*x-4*x^2)])/(2*x^2), {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)) \\ G. C. Greubel, Apr 08 2017

a(n) = Sum_{i=0..floor(n/2)}4^(n-2i)*C(i)*binomial(n-i-1,n), where C(i) is the i-th Catalan number A000108.
G.f.: (1-4*x-sqrt((1-4*x)*(1-4*x-4*x^2)))/(2*x^2).
a(n) ~ 2^(n+3/4) * (1+sqrt(2))^(n+1/2) / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 22 2015
a(n) = ((-16*n + 40)*a(n-3) + (-12*n+12)*a(n-2) +(8*n+4)*a(n-1))/(n+2). - Robert Israel, Apr 22 2015

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

A257363 Number of 3-Motzkin paths with no level steps at height 1.

Original entry on oeis.org

1, 3, 10, 33, 110, 369, 1247, 4248, 14603, 50724, 178314, 635526, 2300829, 8477382, 31842897, 122103276, 478372886, 1915188093, 7831613468, 32674683984, 138871668314, 600140517762, 2631926843602, 11690520554421, 52498671870181, 237966449687118, 1087246253873875, 5001141997115010, 23137102115963262

Offset: 0

José Luis Ramírez Ramírez, Apr 20 2015

nonn

For n=2 we have 10 paths: H(1)H(1), H(1)H(2), H(2)H(1), H(2)H(2), H(1)H(3), H(3)H(1), H(3)H(3), H(2)H(3), H(3)H(2), UD.

Robert Israel, Table of n, a(n) for n = 0..1000

Cf. A117641, A217312, A253831, A059231.

rec:= (95+95*n)*a(n)+(-180-9*n)*a(n+1)+(-329-197*n)*a(n+2)+(369+144*n)*a(n+3)+(-117-36*n)*a(4+n)+(12+3*n)*a(n+5):
f:= gfun:-rectoproc({rec,a(0)=1,a(1)=3,a(2)=10,a(3)=33,a(4)=110},a(n),remember):
seq(f(n),n=0..100); # Robert Israel, Apr 28 2015

CoefficientList[Series[2*(3+x)/(6-17*x-9*x^2+x*Sqrt[1-6*x+5*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 21 2015 *)

G.f.: 1/(1-3*x-x*F(x)), where F(x) is the g.f. of the sequence A117641.
G.f.: 2*(3+x)/(6-17*x-9*x^2+x*sqrt(1-6*x+5*x^2)).
a(n) ~ 5^(n+3/2)/(98*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 21 2015
From Robert Israel, Apr 28 2015 (Start):
G.f.: (6-x*sqrt(1-6*x+5*x^2)-17*x-9*x^2)/(6-36*x+42*x^2+38*x^3).
3*(-n+1)*a(n) +9*(4*n-7)*a(n-1) +9*(-16*n+39)*a(n-2) +(197*n-656)*a(n-3) +9*(n+15)*a(n-4) +95*(-n+4)*a(n-5)=0. (End)

A257517 Number of 3-generalized 2-Motzkin paths of length n with no level steps H=(3,0) at even level.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 5, 8, 18, 30, 66, 120, 252, 484, 1005, 1984, 4110, 8278, 17150, 35024, 72748, 150012, 312642, 649424, 1358244, 2837484, 5954980, 12497616, 26313432, 55434248, 117062205, 247412928, 523881238, 1110335334, 2356819254, 5007428384, 10652412108, 22682131308, 48349084054, 103150869360, 220276819836

Offset: 0

José Luis Ramírez Ramírez, Apr 27 2015

Cf. A257516, A005572, A025266.

CoefficientList[Series[(1-2*x^3-Sqrt[(1-2*x^3)*(1-4*x^2-2*x^3)])/(2*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 27 2015 *)

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

D-finite with recurrence +(n+2)*(n^2-n+3)*a(n) +(n+1)*(n^2+1)*a(n-1) -4*(n-1)*(n^2-n+3)*a(n-2) +2*(-4*n^3+11*n^2-13*n+19)*a(n-3) -2*(2*n-7)*(n^2+1)*a(n-4) +4*(2*n-11)*(n^2-n+3)*a(n-5) +4*(3*n^3-21*n^2+12*n-34)*a(n-6) +4*(n-8)*(n^2+1)*a(n-7)=0. - R. J. Mathar, Jun 07 2016

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

Original entry on oeis.org

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

A252354 Number of Motzkin paths of length n with no level steps at height 2.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 46, 106, 248, 584, 1389, 3329, 8047, 19607, 48167, 119287, 297829, 749632, 1902044, 4864553, 12538933, 32568528, 85224251, 224618900, 596106393, 1592429464, 4280667705, 11575188106, 31474407317, 86029586086, 236292044931, 651952466845

Offset: 0

José Luis Ramírez Ramírez, Apr 20 2015

nonn

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

Cf. A005043, A217312.

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

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

a(n) = a(n-1) + Sum_{j=0..n-2} A217312(j)*a(n-j).
G.f: 1/(1-x-x^2(1/(1-x-x^2*R(x)))), where R(x) is the g.f. of Riordan numbers (A005043).
a(n) ~ 3^(n+3/2) / (32*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 21 2015
Conjecture: (-n+3)*a(n) +3*(2*n-7)*a(n-1) +(-7*n+24)*a(n-2) +2*(-7*n+36)*a(n-3) +2*(11*n-51)*a(n-4) +3*(3*n-23)*a(n-5) +(-10*n+63)*a(n-6) +3*(n-6)*a(n-7)=0. - R. J. Mathar, Sep 24 2016

cp's OEIS Frontend

User: José Luis Ramírez Ramírez

A322378 Triangle read by rows: T(n,k) is the number of nondecreasing Dyck prefixes (i.e., left factors of nondecreasing Dyck paths) of length n and final height k (0 <= k <= n).

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A322329 Triangle read by rows: T(n,k) is the number of nondecreasing Motzkin prefixes (i.e., left factors of nondecreasing Motzkin paths) of length n and final height k (0 <= k <= n).

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A322325 Number of nondecreasing Motzkin paths of length n.

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A253831 Number of 2-Motzkin paths with no level steps at height 1.

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Maxima

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A257390 Number of 4-Motzkin paths of length n with no level steps at even level.

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PARI

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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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Mathematica

Maxima

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A257363 Number of 3-Motzkin paths with no level steps at height 1.

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Maple

Mathematica

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A257517 Number of 3-generalized 2-Motzkin paths of length n with no level steps H=(3,0) at even level.

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