A091866 -id:A091866 - cp's OEIS frontend

A127156 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n starting with exactly k consecutive pyramids. A pyramid in a Dyck path is a factor of the form U^j D^j (j>0), starting at the x-axis. Here U=(1,1) and D=(1,-1). This definition differs from the one in A091866.

1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 5, 2, 3, 3, 1, 19, 7, 5, 6, 4, 1, 67, 26, 12, 11, 10, 5, 1, 232, 93, 38, 23, 21, 15, 6, 1, 804, 325, 131, 61, 44, 36, 21, 7, 1, 2806, 1129, 456, 192, 105, 80, 57, 28, 8, 1, 9878, 3935, 1585, 648, 297, 185, 137, 85, 36, 9, 1, 35072, 13813, 5520

Offset: 0

Emeric Deutsch, Feb 27 2007

Row sums yield the Catalan numbers (A000108). T(n,0)=A114277(n-3) for n>=3. Sum(k*T(n,k), k=0..n)=A014318(n-1) for n>=1.

			T(5,2)=5 because we have (UDUD)UUDUDD, (UUDDUUUDDD), (UUUDDDUUDD), (UDUUUUDDDD) and (UUUUDDDDUD) (the initial 2 pyramids are shown between parentheses).
Triangle starts:
1;
0,1;
0,1,1;
1,1,2,1;
5,2,3,3,1;
19,7,5,6,4,1;

Cf. A000108, A114277, A014318.

C:=(1-sqrt(1-4*z))/2/z: G:=(1-2*z)*C/(1-z-t*z): Gser:=simplify(series(G,z=0,15)): P[0]:=1: for n from 1 to 12 do P[n]:=sort(expand(coeff(Gser,z^n))) od: for n from 0 to 12 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form

G.f.=G(t,z)=(1-2z)C/(1-z-tz), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function. T(n,k)=T(n-1,k)+T(n-1,k-1) for n,k>=1.

A114277 Sum of the lengths of the second ascents in all Dyck paths of semilength n+2.

Original entry on oeis.org

1, 5, 19, 67, 232, 804, 2806, 9878, 35072, 125512, 452388, 1641028, 5986993, 21954973, 80884423, 299233543, 1111219333, 4140813373, 15478839553, 58028869153, 218123355523, 821908275547, 3104046382351, 11747506651599

Offset: 0

Emeric Deutsch, Nov 20 2005

nonn

Also number of Dyck paths of semilength n+4 having length of second ascent equal to three. Example: a(1)=5 because we have UD(UUU)DUDDD, UD(UUU)DDUDD, UD(UUU)DDDUD, UUD(UUU)DDDD and UUDD(UUU)DDD (second ascents shown between parentheses). Partial sums of A002057. Column 3 of A114276. a(n)=absolute value of A104496(n+3).

Also number of Dyck paths of semilength n+3 that do not start with a pyramid (a pyramid in a Dyck path is a factor of the form U^j D^j (j>0), starting at the x-axis; here U=(1,1) and D=(1,-1); this definition differs from the one in A091866). Equivalently, a(n)=A127156(n+3,0). Example: a(1)=5 because we have UUDUDDUD, UUDUDUDD, UUUDUDDD, UUDUUDDD and UUUDDUDD. - Emeric Deutsch, Feb 27 2007

			a(3)=5 because the total length of the second ascents in UD(U)DUD, UD(UU)DD, UUDD(U)D, UUD(U)DD and UUUDDD (shown between parentheses) is 5.

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A002057, A114276, A104496, A127156, A279557.

Cf. A014137 (n=1), A014138 (n=2), A001453 (n=3), this sequence (n=4), A143955 (n=5), A323224 (array).

a:=n->4*sum(binomial(2*j+3,j)/(j+4),j=0..n): seq(a(n),n=0..28);

Table[4*Sum[Binomial[2j+3,j]/(j+4),{j,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 19 2012 *)

from functools import cache
@cache
def B(n, k):
    if n <= 0 or k <= 0: return 0
    if n == k: return 1
    return B(n - 1, k) + B(n, k - 1)
def A114277(n): return B(n + 5, n + 1)
print([A114277(n) for n in range(24)]) # Peter Luschny, May 16 2022

a(n) = 4*Sum_{j=0..n} binomial(2*j+3, j)/(j+4).
G.f.: C^4/(1-z), where C=(1-sqrt(1-4*z))/(2*z) is the Catalan function.
a(n) = c(n+3) - (c(0) + c(1) + ... + c(n+2)), where c(k)=binomial(2k,k)/(k+1) is a Catalan number (A000108). - Emeric Deutsch, Feb 27 2007
D-finite with recurrence: n*(n+4)*a(n) = (5*n^2 + 14*n + 6)*a(n-1) - 2*(n+1)*(2*n+3)*a(n-2). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ 2^(2*n+7)/(3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 19 2012
a(n) = exp((2*i*Pi)/3)-4*binomial(2*n+5,n+1)*hypergeom([1,3+n,n+7/2],[n+2,n+6],4)/ (n+5). - Peter Luschny, Feb 26 2017
a(n-1) = Sum_{i+j+k+lA000108 Catalan number. - Yuchun Ji, Jan 10 2019

More terms from Emeric Deutsch, Feb 27 2007

A109033 Number of permutations in S_n avoiding the patterns 1342 and 2143.

Original entry on oeis.org

1, 1, 2, 6, 22, 88, 368, 1584, 6968, 31192, 141656, 651136, 3023840, 14166496, 66876096, 317809216, 1519163456, 7299577216, 35237444736, 170812433536, 831127053696, 4057858988416, 19873611712896, 97609555091456

Offset: 0

Emeric Deutsch, Jun 16 2005

nonn

Also number of permutations in S_n avoiding the patterns 3142 and 2341. Partial sums of A109034.

Hankel transform is 2^floor(n^2/3) (see A134751). - Paul Barry, Dec 15 2008

			a(4) = 22 because all permutations of 1234 qualify with the exception of 1342 and 2143.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Christian Bean, Finding structure in permutation sets, Ph.D. Dissertation, Reykjavík University, School of Computer Science, 2018.
Christian Bean, Émile Nadeau, and Henning Ulfarsson, Enumeration of Permutation Classes and Weighted Labelled Independent Sets, arXiv:1912.07503 [math.CO], 2019.
Darla Kremer and Wai Chee Shiu, Finite transition matrices for permutations avoiding pairs of length four patterns, Discrete Math. 268 (2003), 171-183. MR1983276 (2004b:05006). See Table 1.
Ian Le, Wilf classes of pairs of permutations of length 4, Electron. J. Combin., 12(1) (2005), R25, 26 pages.
E. Rowland and R. Yassawi, Automatic congruences for diagonals of rational functions, arXiv preprint arXiv:1310.8635 [math.NT], 2013-2014.
Wikipedia, Permutation classes avoiding two patterns of length 4.

Cf. A109034.

G:=(1-sqrt(1-8*x+16*x^2-8*x^3))/4/x/(1-x): Gser:=series(G,x=0,30): 1,seq(coeff(Gser,x^n),n=1..27);

CoefficientList[Series[(1-Sqrt[1-8x+16x^2-8x^3])/(4x(1-x)), {x,0,30}], x] (* Harvey P. Dale, Jul 02 2011 *)

G.f.: (1-sqrt(1-8*x+16*x^2-8*x^3))/(4*x*(1-x)).
From Paul Barry, Dec 15 2008: (Start)
G.f.: (1-x)*c(2*x*(1-x)^2), where c(x) is the g.f. of A000108;
a(n) = sum{k=0..n, (-1)^(n-k)*C(2k+1,n-k)*2^k*A000108(k)}. (End)
G.f.: 1/(1-x/(1-x/(1-2x/(1-x/(1-x/(1-2x/(1-x/(1-x/(1-2x...... (continued fraction). - Paul Barry, Dec 15 2008
a(n) = Sum_{k=0..n} A091866(n,k)*2^(n-k). - Philippe Deléham, Nov 27 2009
Recurrence: (n+1)*a(n) = 3*(3*n-1)*a(n-1) - 12*(2*n-3)*a(n-2) + 12*(2*n-5) * a(n-3) - 4*(2*n-7)*a(n-4). - Vaclav Kotesovec, Oct 24 2012
a(n) ~ sqrt(5-sqrt(5))*(sqrt(5)+3)^n/(2*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 24 2012. Equivalently, a(n) ~ 5^(1/4) * 2^(n-1) * phi^(2*n - 1/2) / (sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 10 2021
G.f. A(x) satisfies: A(x) = (1 - x) * (1 + 2*x*A(x)^2). - Ilya Gutkovskiy, Jun 30 2020

A121462 Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n, having pyramid weight k (1 <= k <= n).

Original entry on oeis.org

1, 0, 2, 0, 1, 4, 0, 1, 4, 8, 0, 1, 5, 12, 16, 0, 1, 6, 18, 32, 32, 0, 1, 7, 25, 56, 80, 64, 0, 1, 8, 33, 88, 160, 192, 128, 0, 1, 9, 42, 129, 280, 432, 448, 256, 0, 1, 10, 52, 180, 450, 832, 1120, 1024, 512, 0, 1, 11, 63, 242, 681, 1452, 2352, 2816, 2304, 1024, 0, 1, 12, 75, 316

Offset: 1

Emeric Deutsch, Jul 31 2006

A pyramid in a Dyck word (path) is a factor of the form U^h D^h, where U=(1,1), D=(1,-1) and h is the height of the pyramid. A pyramid in a Dyck word w is maximal if, as a factor in w, it is not immediately preceded by a u and immediately followed by a d. The pyramid weight of a Dyck path (word) is the sum of the heights of its maximal pyramids.
Row sums are the odd-subscripted Fibonacci numbers (A001519). T(n,n)=2^(n-1). Sum_{k=1..n} k*T(n,k) = A030267(n).
Mirror image of triangle in A153342. - Philippe Deléham, Dec 31 2008
Essentially triangle given by (0,1/2,1/2,0,0,0,0,0,0,0,...) DELTA (2,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 30 2011
A121462 is jointly generated with A208341 as an array of coefficients of polynomials u(n,x):  initially, u(1,x)=v(1,x)=1; for n > 1, u(n,x) = x*u(n-1,x) + x*v(n-1) and v(n,x) = x*u(n-1,x) + (x+1)*v(n-1,x).  See the Mathematica section. - Clark Kimberling, Mar 11 2012

			T(4,3)=4 because we have (UD)U(UD)(UD)D, U(UD)(UD)(UD)D, U(UD)(UUDD)D and U(UUDD)(UD)D, where U=(1,1) and D=(1,-1) (the maximal pyramids are shown between parentheses).
Triangle starts:
  1;
  0,  2;
  0,  1,  4;
  0,  1,  4,  8;
  0,  1,  5, 12, 16;
  0,  1,  6, 18, 32, 32;

Elena Barcucci, Alberto del Lungo, S. Fezzi and Renzo Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
Nicolas Bělohoubek and Antonín Slavík, L-Tetromino Tilings and Two-Color Integer Compositions, Univ. Karlova (Czechia, 2025). See p. 10.
Alain Denise and Rodica Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155-176.
Emeric Deutsch and Helmut Prodinger, A bijection between directed column-convex polyominoes and ordered trees of height at most three, Theoretical Comp. Science, 307, 2003, 319-325.

Cf. A001519, A030267, A091866.

T:=proc(n,k) if n=1 and k=1 then 1 elif k=1 then 0 elif k<=n then sum(binomial(k-1,j)*binomial(n-k-1+j,j-1),j=0..k-1) else 0 fi end: for n from 1 to 13 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + (x + 1) v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A121462 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208341 *)
(* Clark Kimberling, Mar 11 2012 *)

T(n,k) = Sum_{j=0..k-1} binomial(k-1,j)*binomial(n-k-1+j,j-1) for 2 <= k <= n; T(1,1)=1; T(n,1)=0 for n >= 2.
G.f.: G = G(t,z) = tz(1-z)/(1-2tz-z+tz^2).
T(n+1,k+1) = A062110(n,k)*2^(2*k-n). - Philippe Deléham, Aug 01 2006

A186338 Expansion of 1/(1-2x/(1-2x/(1-x/(1-2x/(1-2x/(1-x/(1-2x/(1-... (continued fraction).

Original entry on oeis.org

1, 2, 8, 36, 172, 860, 4460, 23820, 130268, 726236, 4112972, 23599724, 136906748, 801671996, 4732110828, 28128179276, 168222049052, 1011509012636, 6111445499532, 37084090264364, 225899543897916, 1380918157453052, 8468524718133804, 52085281291575052

Offset: 0

Paul Barry, Feb 18 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Xiaomei Chen, Yuan Xiang, Counting generalized Schröder paths, arXiv:2009.04900 [math.CO], 2020.

Hankel transform is A186339.

CoefficientList[Series[(Sqrt[1-10*x+25*x^2-16*x^3]+3*x-1)/(2*x*(2*x-1)), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 24 2012 *)

a(n):=sum(sum(binomial(j+1, i)*binomial(2*j-i, j-i)*binomial(n-j+i-1, n-j), i, 0, j)/(j+1)*2^(n-j), j, 0, n); /* Vladimir Kruchinin, Jan 25 2020 */

G.f.: (sqrt(1-10x+25x^2-16x^3)+3x-1)/(2x(2x-1)).
Conjecture: (n+1)*a(n) +3*(1-4n)*a(n-1) +15*(3n-4)*a(n-2) +6*(26-11n)*a(n-3) +16*(2n-7)*a(n-4)=0. - R. J. Mathar, Nov 17 2011
a(n) = Sum_{k, 0<=k<=n} A091866(n,k)*2^k. - Philippe Deléham, Nov 27 2011
a(n) ~ sqrt(7*sqrt(17)-17)*((9+sqrt(17))/2)^n/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 24 2012
From Vladimir Kruchinin, Jan 25 2020: (Start)
a(n) = Sum_{j=0..n} Sum_{i=0..j} C(j+1, i)*C(2*j-i, j-i)*C(n-j+i-1,n-j) /(j+1)*2^(n-j).
a(n) = Sum_{i=0..n-1} a(i)*(2^(n-i-1)+a(n-i-1)). (End)

A166302 Sum of pyramid weights of all Dyck paths of semilength n that have no ascents and no descents of length 1.

Original entry on oeis.org

0, 0, 2, 3, 8, 19, 44, 106, 257, 628, 1549, 3844, 9588, 24020, 60391, 152298, 385085, 975904, 2478129, 6303861, 16060946, 40977605, 104682165, 267730426, 685451776, 1756593392, 4505537267, 11565724164, 29711413595, 76379060176, 196473781247

Offset: 0

Emeric Deutsch, Nov 07 2009

nonn

A pyramid in a Dyck word (path) is a factor of the form U^h D^h, h being the height of the pyramid. A pyramid in a Dyck word w is maximal if, as a factor in w, it is not immediately preceded by a U and immediately followed by a D. The pyramid weight of a Dyck path (word) is the sum of the heights of its maximal pyramids.

			a(5)=19 because the pyramid weights of (UUDD)(UUUDDD), (UUUDDD)(UUDD), U(UUDD)(UUDD)D, and (UUUUUDDDDD) are 5, 5, 4, and 5, respectively (the maximal pyramids are shown between parentheses).

G. C. Greubel, Table of n, a(n) for n = 0..1000
A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155-176.

Cf. A166301, A091866.

G := (1/2)*z*(2-z)*(1+z-z^2-sqrt((1+z+z^2)*(1-3*z+z^2)))/((1-z)*sqrt((1+z+z^2)*(1-3*z+z^2))): Gser := series(G, z = 0, 35): seq(coeff(Gser, z, n), n = 0 .. 32);

CoefficientList[Series[1/2*x*(2-x)*(1+x-x^2-Sqrt[(1+x+x^2)*(1-3*x+x^2)]) /((1-x)*Sqrt[(1+x+x^2)*(1-3*x+x^2)]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)

z='z+O('z^50); concat([0,0], Vec(z*(2-z)*(1+z-z^2-sqrt((1+z+z^2)*(1-3*z+z^2)))/(2*(1-z)*sqrt((1+z+z^2)*(1-3*z+z^2))))) \\ G. C. Greubel, Mar 22 2017

a(n) = Sum_{k=0..n} k*A166301(n,k).
G.f.: z*(2-z)*[1+z-z^2-sqrt((1+z+z^2)*(1-3*z+z^2))]/[2*(1-z)*sqrt((1+z+z^2)*(1-3*z+z^2))].
a(n) ~ (3+sqrt(5))^(n+1/2) / (5^(1/4) * sqrt(Pi*n) * 2^(n+3/2)). - Vaclav Kotesovec, Mar 20 2014
D-finite with recurrence 2*(n-1)*(2494*n-8185)*a(n) +23*(-770*n^2+3867*n-3959)*a(n-1) +(13226*n^2-83741*n+101091)*a(n-2) +(-7734*n^2+51213*n-51521)*a(n-3) +(17710*n^2-114385*n+144471)*a(n-4) +(-13226*n^2+104701*n-162397)*a(n-5) +(2746*n-6149)*(n-7)*a(n-6)=0. - R. J. Mathar, Jul 24 2022

A168494 Sequence with Hankel transform equal to 3^floor(n^2/3).

Original entry on oeis.org

1, 1, 2, 7, 32, 160, 830, 4405, 23798, 130498, 724748, 4069258, 23064608, 131809108, 758696492, 4394825647, 25600773272, 149877922228, 881394158558, 5204245242208, 30841413359186, 183381577399006, 1093695670905206

Offset: 0

Paul Barry, Nov 27 2009

Hankel transform is A168495.

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A000108, A091866, A109033. - Philippe Deléham, Nov 27 2009

CoefficientList[Series[(1+x-Sqrt[1-10*x+25*x^2-12*x^3])/(6*x*(1-x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 24 2012 *)

G.f.: 1/(1-x/(1-x/(1-3x/(1-x/(1-x/(1-3x/(1-x/(1-x/(1-3x/(1-.... (continued fraction);
G.f.: 1/(1-x-x^2/(1-4x-3x^2/(1-2x-3x^2/(1-4x-x^2/(1-4x-3x^2/(1-2x-3x^2/(1-4x-x^2/(1-... (continued fraction),
with sequences (1,3,3,1,3,3,1,3,3,1,...) and (1,4,2,4,4,2,4,4,2,4,4,...).
G.f.: (1+x-sqrt(1-10x+25x^2-12x^3))/(6x(1-x)).
a(n) = Sum_{k=0..n} A091866(n,k)*3^(n-k). - Philippe Deléham, Nov 27 2009
Conjecture: (n+1)*a(n) +(4-11*n)*a(n-2) +5*(7*n-11)*a(n-2) +(92-37*n) * a(n-3) +6*(2*n-7)*a(n-4) = 0. - R. J. Mathar, Sep 30 2012
a(n) ~ sqrt(33-sqrt(33))*((7+sqrt(33))/2)^n/(12*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 24 2012

A174015 A generalized Catalan number sequence.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 2, -3, -10, -14, -4, 34, 104, 172, 132, -197, -964, -1976, -2190, 652, 9294, 23626, 33762, 12140, -84438, -280850, -493930, -397666, 639678, 3248466, 6947462, 8068589, -2165978, -35591958, -94129444, -139864826, -56393480, 352505724

Offset: 0

Paul Barry, Mar 05 2010

Hankel transform is A130151(n+1). First column of A174014.

G.f.: (sqrt(1-2x+x^2+4x^3)+3x-1)/(2x(1-x));
G.f.: 1/(1-x/(1-x/(1+x/(1-x/(1-x/(1+x/(1-... (continued fraction).
a(n) = Sum_{k, 0<=k<=n} A091866(n,k)*(-1)^(n-k) = Sum_{k, 0<=k<=n} A198379(n,k). - Philippe Deléham, Nov 27 2011
Conjecture: (n+1)*a(n) -3*n*a(n-1) +3*(n-1)*a(n-2) +3*(n-4)*a(n-3) +2*(7-2*n)*a(n-4)=0. R. J. Mathar, Nov 13 2012

More terms from Philippe Deléham, Oct 27 2011

A094322 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k pyramids.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 4, 3, 3, 3, 1, 13, 11, 7, 6, 4, 1, 42, 37, 23, 14, 10, 5, 1, 139, 122, 78, 43, 25, 15, 6, 1, 470, 408, 262, 145, 75, 41, 21, 7, 1, 1616, 1390, 887, 494, 251, 124, 63, 28, 8, 1, 5632, 4810, 3048, 1694, 864, 414, 196, 92, 36, 9, 1, 19852, 16857, 10622

Offset: 0

Emeric Deutsch, Jun 03 2004

A pyramid in a Dyck path is a factor of the form U^j D^j (j>0), starting at the x-axis. Here U=(1,1) and D(1,-1). This definition differs from the one in A091866. Column k=0 is A082989. Row sums are the Catalan numbers (A000108).

			T(3,2)=2 because there are two Dyck paths of semilength 3 having 2 pyramids: (UD)(UUDD) and (UUDD)(UD) (pyramids shown between parentheses).
Triangle begins:
[1];
[0, 1];
[0, 1, 1];
[1, 1, 2, 1];
[4, 3, 3, 3, 1];
[13, 11, 7, 6, 4, 1];
[42, 37, 23, 14, 10, 5, 1];

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A082989, A000108.

C:=(1-sqrt(1-4*z))/2/z: G:=(1-z)/(1-z*C+z^2*C-t*z): Gserz:=simplify(series(G,z=0,16)): P[0]:=1: for n from 1 to 14 do P[n]:=sort(coeff(Gserz,z^n)) od: seq([subs(t=0,P[n]),seq(coeff(P[n],t^k),k=1..n)],n=0..14);
# second Maple program:
b:= proc(x, y, u, t) option remember; expand(`if`(y<0 or y>x, 0,
      `if`(x=0, `if`(t, z, 1), (b(x-1, y-1, false, t)+
      b(x-1, y+1, true, t and u or y=0))*`if`(t and y=0, z, 1))))
    end:
T:= n-> (p-> seq(coeff(p,z,i), i=0..n))(b(2*n, 0, false$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Jul 22 2015

b[x_, y_, u_, t_] := b[x, y, u, t] = Expand[If[y<0 || y>x, 0, If[x==0, If[ t, z, 1], (b[x-1, y-1, False, t] + b[x-1, y+1, True, t && u || y == 0]) * If[t && y==0, z, 1]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, n}]][b[2*n, 0, False, False]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 29 2016, after Alois P. Heinz *)

G.f.: G=G(t,z) = (1-z)/(1-zC+z^2*C -tz), where C = [1-sqrt(1-4z)]/(2z) is the Catalan function.

A129163 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and pyramid weight k.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 4, 11, 13, 8, 8, 29, 46, 38, 16, 16, 74, 150, 167, 104, 32, 32, 184, 461, 652, 554, 272, 64, 64, 448, 1354, 2344, 2535, 1724, 688, 128, 128, 1072, 3836, 7922, 10462, 9103, 5112, 1696, 256, 256, 2528, 10552, 25506, 40007, 42547, 30773, 14592

Offset: 1

Emeric Deutsch, Apr 03 2007

A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. A pyramid in a skew Dyck word (path) is a factor of the form U^h D^h, h being the height of the pyramid. A pyramid in a skew Dyck word w is maximal if, as a factor in w, it is not immediately preceded by a U and immediately followed by a D. The pyramid weight of a skew Dyck path (word) is the sum of the heights of its maximal pyramids.

Row sums yield A002212. T(n,1)=2^(n-2) (n>=2). T(n,n)=2^(n-1). Sum(k*T(n,k),k=1..n)=A129164(n). Pyramid weight in Dyck paths is considered in the Denise and Simion reference (see also A091866).

			T(3,2)=4 because we have (UD)U(UD)L, U(UD)(UD)D, U(UD)(UD)L and U(UUDD)L (the maximal pyramids are shown between parentheses).
Triangle starts:
1;
1,2;
2,4,4;
4,11,13,8;
8,29,46,38,16;

A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155-176.
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203

Cf. A002212, A129164.

eq:=z*(1-t*z)*G^2-(1-2*t*z+t*z^2)*G+(1-z)*(1-t*z)=0: G:=RootOf(eq,G): Gser:=simplify(series(G-1,z=0,15)): for n from 1 to 11 do P[n]:=sort(expand(coeff(Gser,z,n))) od: for n from 1 to 11 do seq(coeff(P[n],t,j),j=1..n) od; # yields sequence in triangular form

G.f.=G-1, where G=G(t,z) is given by z(1-tz)G^2-(1-2tz+tz^2)G+(1-z)(1-tz)=0.

cp's OEIS Frontend

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