A114848 -id:A114848 - cp's OEIS frontend

A243752 Number T(n,k) of Dyck paths of semilength n having exactly k (possibly overlapping) occurrences of the consecutive step pattern given by the binary expansion of n, where 1=U=(1,1) and 0=D=(1,-1); triangle T(n,k), n>=0, read by rows.

1, 0, 1, 0, 1, 1, 1, 3, 1, 1, 11, 2, 9, 16, 12, 4, 1, 1, 57, 69, 5, 127, 161, 98, 35, 7, 1, 323, 927, 180, 1515, 1997, 1056, 280, 14, 4191, 5539, 3967, 1991, 781, 244, 64, 17, 1, 1, 10455, 25638, 18357, 4115, 220, 1, 20705, 68850, 77685, 34840, 5685, 246, 1

Offset: 0

Alois P. Heinz, Jun 09 2014

			Triangle T(n,k) begins:
: n\k :    0     1     2     3    4    5  ...
+-----+----------------------------------------------------------
:  0  :    1;                                 [row  0 of A131427]
:  1  :    0,    1;                           [row  1 of A131427]
:  2  :    0,    1,    1;                     [row  2 of A090181]
:  3  :    1,    3,    1;                     [row  3 of A001263]
:  4  :    1,   11,    2;                     [row  4 of A091156]
:  5  :    9,   16,   12,    4,   1;          [row  5 of A091869]
:  6  :    1,   57,   69,    5;               [row  6 of A091156]
:  7  :  127,  161,   98,   35,   7,   1;     [row  7 of A092107]
:  8  :  323,  927,  180;                     [row  8 of A091958]
:  9  : 1515, 1997, 1056,  280,  14;          [row  9 of A135306]
: 10  : 4191, 5539, 3967, 1991, 781, 244, ... [row 10 of A094507]

Alois P. Heinz, Rows n = 0..270, flattened

Columns k=0-10 give: A243754, A243770, A243771, A243772, A243773, A243774, A243775, A243776, A243777, A243778, A243779, or main diagonals of A243753, A243827, A243828, A243829, A243830, A243831, A243832, A243833, A243834, A243835, A243836.
Row sums give A000108.
Cf. A098978, A114463, A114848, A116424, A135305, A242450, A243366, A243838.

A187256 Number of peakless Motzkin paths of length n, assuming that the (1,0)-steps come in 2 colors.

Original entry on oeis.org

1, 2, 4, 10, 28, 82, 248, 770, 2440, 7858, 25644, 84618, 281844, 946338, 3199728, 10885122, 37230352, 127951714, 441633812, 1530242954, 5320853260, 18560408050, 64932101224, 227767796482, 800928670232, 2822814469394, 9969770245948, 35280714655498

Offset: 0

Emeric Deutsch, May 03 2011

nonn

Ordinary peakless Motzkin paths are counted by A004148.

Also the number of Catalan words of length n avoiding the consecutive pattern 010. - Sela Fried, May 21 2025

			a(4)=28 because, denoting U=(1,1), D=(1,-1), and H=(1,0), we have 2^4=16 paths of shape HHHH, 2^2=4 paths of shape HUHD, 2^2 = 4 paths of shape UHDH, and 4 paths of shape UHHD.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv:1107.5490 [math.CO], 2011.
Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.

Cf. A004148, A328504

Column k=0 of A114848 (shifted). - Alois P. Heinz, Mar 31 2016

eq := G = 1+2*z*G+z^2*G*(G-1): G := RootOf(eq, G): Gser := series(G, z = 0, 30): seq(coeff(Gser, z, n), n = 0 .. 27);

CoefficientList[Series[(1 + (x-2)*x - Sqrt[(1 + (x-4)*x)*(1+x^2)])/(2*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 02 2014 *)
a[n_] := 2^n HypergeometricPFQ[{-n/2, (1 - n)/2, (1 - n)/2, 1 - n/2}, {2, -n, -n + 1}, 4]; Array[a, 28, 0] (* Peter Luschny, Jan 25 2020 *)

a(n):=sum(((-1)^i*binomial(n-i,i)*binomial(2*n-4*i+2,n-2*i))/(n-2*i+1),i,0,(n)/2); /* Vladimir Kruchinin, Jun 01 2014 */

my(x='x+O('x^50)); Vec((1 + (x-2)*x - sqrt((1 + (x-4)*x)*(1+x^2))) /( 2*x^2)) \\ G. C. Greubel, Feb 12 2017

G.f.: G(z) satisfies the equation G = 1 + 2*z*G + z^2*G*(G-1).
Conjecture: (n+2)*a(n) -2*(2*n+1)*a(n-1) +2*(n-1)*a(n-2) +2*(5-2*n)*a(n-3) +(n-4)*a(n-4) = 0. - R. J. Mathar, Nov 16 2011
a(n) = Sum_{i=0..n/2} (-1)^i*binomial(n-i,i)*binomial(2*n-4*i+2,n-2*i)/(n-2*i+1). - Vladimir Kruchinin, Jun 01 2014
a(n) ~ sqrt(24+14*sqrt(3)) * (2+sqrt(3))^n / (2*sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jun 02 2014
a(n) = 2^n*hypergeom([-n/2, (1 - n)/2, (1 - n)/2, 1 - n/2], [2, -n, -n + 1], 4). - Peter Luschny, Jan 25 2020

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A243752 Number T(n,k) of Dyck paths of semilength n having exactly k (possibly overlapping) occurrences of the consecutive step pattern given by the binary expansion of n, where 1=U=(1,1) and 0=D=(1,-1); triangle T(n,k), n>=0, read by rows.

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A187256 Number of peakless Motzkin paths of length n, assuming that the (1,0)-steps come in 2 colors.

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