A091869 -id:A091869 - 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.

A097692 Triangle read by rows: a(n,k) = number of paths of n upsteps U and n downsteps D that contain k UDUs.

Original entry on oeis.org

1, 2, 4, 2, 10, 8, 2, 26, 30, 12, 2, 70, 104, 60, 16, 2, 192, 350, 260, 100, 20, 2, 534, 1152, 1050, 520, 150, 24, 2, 1500, 3738, 4032, 2450, 910, 210, 28, 2, 4246, 12000, 14952, 10752, 4900, 1456, 280, 32, 2, 12092, 38214, 54000, 44856, 24192, 8820, 2184, 360, 36, 2

Offset: 0

David Callan, Aug 19 2004; corrected Jun 10 2005

See A091869 for the distribution of the parameter "number of UDUs" on Dyck paths.

			Table begins
\ k 0, 1, 2, ...
n
0 | 1
1 | 2
2 | 4, 2
3 | 10, 8, 2
4 | 26, 30, 12, 2
5 | 70, 104, 60, 16, 2
6 |192, 350, 260, 100, 20, 2
7 |534, 1152, 1050, 520, 150, 24, 2
The path UDUDUD contains 2 UDUs and a(2,1) = 2 because each of UDUD, DUDU contains one UDU.

Aristidis Sapounakis, Panagiotis Tsikouras, Ioannis Tasoulas, Kostas Manes, Strings of Length 3 in Grand-Dyck Paths and the Chung-Feller Property, Electr. J. Combinatorics, 19 (2012), #P2. - From N. J. A. Sloane, Feb 06 2013

Alois P. Heinz, Rows n = 0..141, flattened

Column k=0 is A025565. The row sums are the (even) central binomial coefficients A000984.

Cf. A171651.

b:= proc(u, d, t) option remember; `if`(u=0 and d=0, 1,
      expand(`if`(u=0, 0, b(u-1, d, 2)*`if`(t=3, x, 1))
      +`if`(d=0, 0, b(u, d-1, `if`(t=2, 3, 1)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 1)):
seq(T(n), n=0..12);  # Alois P. Heinz, Apr 29 2015

gfForBalancedByNumberUDU=Sqrt[(1 + x - x*y)/(1 - 3*x - x*y)]; Map[CoefficientList[ #, y]&, CoefficientList[Normal[Series[gfForBalancedByNumberUDU, {x, 0, 8}, {y, 0, 8}]], x]]

G.f.: ((1 + x - x*y)/(1 - 3*x - x*y))^(1/2) = Sum_{n>=0, k>=0} a(n,k) x^n y^k.

Keyword tabl changed to tabf by Michel Marcus, Apr 07 2013

A091187 Triangle read by rows: T(n,k) is the number of ordered trees with n edges and k branches.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 4, 1, 4, 12, 16, 9, 1, 5, 20, 40, 45, 21, 1, 6, 30, 80, 135, 126, 51, 1, 7, 42, 140, 315, 441, 357, 127, 1, 8, 56, 224, 630, 1176, 1428, 1016, 323, 1, 9, 72, 336, 1134, 2646, 4284, 4572, 2907, 835, 1, 10, 90, 480, 1890, 5292, 10710, 15240, 14535, 8350, 2188

Offset: 1

Emeric Deutsch, Feb 23 2004

Row sums are the Catalan numbers A000108. Diagonal entries are the Motzkin numbers A001006.
Equals binomial transform of an infinite lower triangular matrix with A001006 as the main diagonal and the rest zeros. [Gary W. Adamson, Dec 31 2008] [Corrected by Paul Barry, Mar 06 2011]
Reversal of A091869. Diagonal sums are A026418(n+2). [Paul Barry, Mar 06 2011]

			Triangle begins:
  1;
  1, 1;
  1, 2,  2;
  1, 3,  6,   4;
  1, 4, 12,  16,   9;
  1, 5, 20,  40,  45,  21;
  1, 6, 30,  80, 135, 126,  51;
  1, 7, 42, 140, 315, 441, 357, 127;

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1..150, flattened)
J.-L. Baril, S. Kirgizov, The pure descent statistic on permutations, Preprint, 2016.
J. Riordan, Enumeration of plane trees by branches and endpoints, J. Combinat. Theory, Ser A, 19, 214-222, 1975.
Lin Yang and Shengliang Yang, Protected Branches in Ordered Trees, J. Math. Study (2023) Vol. 56, No. 1, 1-17.

Cf. A001006, A000108.

Cf. A007476. [Gary W. Adamson, Dec 31 2008]

M := n->sum(binomial(n+1,q)*binomial(n+1-q,q-1),q=0..ceil((n+1)/2))/(n+1): T := (n,k)->binomial(n-1,k-1)*M(k-1): seq(seq(T(n,k),k=1..n),n=1..13);

(* m = MotzkinNumber *) m[0] = 1; m[n_] := m[n] = m[n - 1] + Sum[m[k]*m[n - 2 - k], {k, 0, n - 2}]; t[n_, k_] := m[k - 1]*Binomial[n - 1, k - 1]; Table[t[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 10 2013 *)

T(n,k) = M(k-1)*binomial(n-1, k-1), where M(k) = A001006(k) = (Sum_{q=0..ceiling((k+1)/2)} binomial(k+1, q)*binomial(k+1-q, q-1))/(k+1) is a Motzkin number.
G.f.: G = G(t,z) satisfies t*z*G^2 -(1 - z + t*z)*G + 1- z + t*z = 0.
From Paul Barry, Mar 06 2011: (Start)
G.f.: 1/(1-x-xy-x^2y^2/(1-x-xy-x^2y^2/(1-x-xy-x^2y^2/(1-... (continued fraction).
G.f.: (1-x(1+y)-sqrt(1-2x(1+y)+x^2(1+2y-3y^2)))/(2x^2*y^2).
E.g.f.: exp(x(1+y))*Bessel_I(1,2*x*y)/(x*y). (End)

A371408 Number of Dyck paths of semilength n having exactly three (possibly overlapping) occurrences of the consecutive step pattern UDU, where U = (1,1) and D = (1,-1).

Original entry on oeis.org

0, 0, 0, 0, 1, 4, 20, 80, 315, 1176, 4284, 15240, 53295, 183700, 625768, 2110472, 7057505, 23427600, 77271120, 253426752, 827009523, 2686728060, 8693388060, 28026897360, 90058925649, 288516259416, 921755412900, 2937377079000, 9338728806225, 29626186593276

Offset: 0

Alois P. Heinz, Mar 22 2024

nonn

			a(4) = 1: UDUDUDUD.
a(5) = 4: UDUDUDUUDD, UDUDUUDUDD, UDUUDUDUDD, UUDUDUDUDD.

Alois P. Heinz, Table of n, a(n) for n = 0..2090
Wikipedia, Counting lattice paths

Column k=3 of A091869.

Cf. A000108, A001006, A005717, A102839, A121262.

a:= n-> `if`(n<4, 0, binomial(n-1, 3)*add(binomial(n-3, j)*
         binomial(n-3-j, j-1), j=0..ceil((n-3)/2))/(n-3)):
seq(a(n), n=0..29);
# second Maple program:
a:= proc(n) option remember; `if`(n<5, [0$4, 1][n+1],
     (n-1)*((2*n-7)*a(n-1)+3*(n-2)*a(n-2))/((n-2)*(n-4)))
    end:
seq(a(n), n=0..29);

a(n) mod 2 = A121262(n) for n >= 1.

A375253 Expansion of (1 - 2x + 2x^2)/(1 - 2x - 3x^2)^(7/2).

Original entry on oeis.org

1, 5, 30, 140, 630, 2646, 10710, 41910, 159885, 597025, 2190188, 7914270, 28230020, 99567300, 347720040, 1203777072, 4135047615, 14105322315, 47813634330, 161154659820, 540353553894, 1803226621350, 5991410183850, 19827295283250, 65371101643575

Offset: 0

Seiichi Manyama, Aug 07 2024

nonn

Column k=4 of A091869 (with a different offset).
Cf. A374506, A375248.
Cf. A005717, A373651.

a[n_]:=(1+n)(2+n)(3+n)(4+n)Hypergeometric2F1[(1-n)/2,-n/2,2,4]/24; Array[a,25,0] (* Stefano Spezia, Aug 07 2024 *)

my(N=30, x='x+O('x^N)); Vec((1-2*x+2*x^2)/(1-2*x-3*x^2)^(7/2))

a(n) = (binomial(n+4,3)/4) * Sum_{k=0..floor(n/2)} binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = (binomial(n+4,3)/4) * A005717(n+1).
a(n) = ((n+4)/(n*(n+2))) * ((2*n+1)*a(n-1) + 3*(n+3)*a(n-2)).
a(n) = (1 + n)*(2 + n)*(3 + n)*(4 + n)*hypergeom([(1-n)/2, -n/2], [2], 4)/24. - Stefano Spezia, Aug 07 2024

A375259 Expansion of (1 - 3x + 6x^2 - 4x^3)/(1 - 2x - 3*x^2)^(9/2).

Original entry on oeis.org

1, 6, 42, 224, 1134, 5292, 23562, 100584, 415701, 1671670, 6570564, 25325664, 95982068, 358442280, 1321336152, 4815108288, 17367199983, 62063418186, 219942717918, 773542367136, 2701767769470, 9376778431020, 32353614992790, 111032853586200, 379152389532735

Offset: 0

Seiichi Manyama, Aug 08 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Column k=5 of A091869 (with a different offset).

Cf. A005717, A375260.

my(N=30, x='x+O('x^N)); Vec((1-3*x+6*x^2-4*x^3)/(1-2*x-3*x^2)^(9/2))

a(n) = (binomial(n+5,4)/5) * Sum_{k=0..floor(n/2)} binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = (binomial(n+5,4)/5) * A005717(n+1).
a(n) = ((n+5)/(n*(n+2))) * ((2*n+1)*a(n-1) + 3*(n+4)*a(n-2)).

A371411 Number of Dyck paths of semilength 2n having exactly n (possibly overlapping) occurrences of the consecutive step pattern UDU, where U = (1,1) and D = (1,-1).

Original entry on oeis.org

1, 1, 3, 20, 140, 1134, 9702, 87516, 817245, 7852130, 77135630, 771742608, 7839348244, 80661853300, 839138980500, 8813312133840, 93339369441540, 995827949882370, 10694044148599350, 115515073043785800, 1254354063204682440, 13685749828961247180

Offset: 0

Alois P. Heinz, Mar 22 2024

nonn

			a(1) = 1: UDUD.
a(2) = 3: UDUDUUDD, UDUUDUDD, UUDUDUDD.
a(3) = 20: UDUDUDUUDDUD, UDUDUDUUUDDD, UDUDUUDDUDUD, UDUDUUDUDDUD, UDUDUUDUUDDD, UDUDUUUDUDDD, UDUUDDUDUDUD, UDUUDUDDUDUD, UDUUDUDUDDUD, UDUUDUDUUDDD, UDUUDUUDUDDD, UDUUUDUDUDDD, UUDDUDUDUDUD, UUDUDDUDUDUD, UUDUDUDDUDUD, UUDUDUDUDDUD, UUDUDUDUUDDD, UUDUDUUDUDDD, UUDUUDUDUDDD, UUUDUDUDUDDD.
a(4) = 140: UDUDUDUDUUDDUUDD, UDUDUDUDUUUDDDUD, UDUDUDUDUUUDDUDD, ..., UUUDUDUUDUDUDDDD, UUUDUUDUDUDUDDDD, UUUUDUDUDUDUDDDD.

Alois P. Heinz, Table of n, a(n) for n = 0..932
Wikipedia, Counting lattice paths

Cf. A000108, A081294, A091869.

a:= proc(n) option remember; `if`(n<2, 1, (2*(n-1)*(2*n-1)^2*
      a(n-1)+12*(n-2)*(2*n-1)*(2*n-3)*a(n-2))/((n+1)*n*(n-1)))
    end:
seq(a(n), n=0..21);

a(n) = A091869(2n,n).

a(n) mod 2 = 1 <=> n in { round(2^(2*k-3)) : k >= 0 } = { A081294 } U { 0 }.

A098747 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having exactly k UDU's at low level.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 8, 4, 1, 1, 24, 11, 5, 1, 1, 75, 35, 14, 6, 1, 1, 243, 113, 47, 17, 7, 1, 1, 808, 376, 156, 60, 20, 8, 1, 1, 2742, 1276, 532, 204, 74, 23, 9, 1, 1, 9458, 4402, 1840, 712, 257, 89, 26, 10, 1, 1, 33062, 15390, 6448, 2507, 917, 315, 105, 29, 11, 1, 1, 116868

Offset: 1

N. J. A. Sloane, Oct 30 2004

T(n,0) = A000958(n-1). - Emeric Deutsch, Dec 23 2006

			Triangle begins:
1
1 1
3 1 1
8 4 1 1
24 11 5 1 1
75 35 14 6 1 1
T(4,2)=1 because we have UDUDUUDD.

Yidong Sun, The statistic "number of udu's" in Dyck paths, Discrete Math., 287 (2004), 177-186.

Cf. A091869, A092107.

Cf. A000958.

c:=(1-sqrt(1-4*z))/2/z: G:=z*c/(1-t*z+z-z*c): Gser:=simplify(series(G,z=0,15)): for n from 1 to 13 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 1 to 12 do seq(coeff(P[n],t,k),k=0..n-1) od; # yields sequence in triangular form - Emeric Deutsch, Dec 23 2006

u[n_, k_, i_]:=(2i+1)/(n-k)Binomial[k+i, i]Binomial[2n-2k-2i-2, n-k-1] u[n_, k_]/;k<=n-1 := Sum[u[n, k, i], {i, 0, n-k-1}] Table[u[n, k], {n, 10}, {k, 0, n-1}] (* u[n, k, i] is the number of Dyck n-paths with k low UDUs and k+i+1 returns altogether. For example, with n=4, k=1 and i=1, u[n, k, i] counts UDUUDDUD, UUDDUDUD because each has size n=4, k=1 low UDUs and k+i+1=3 returns to ground level. *) (* David Callan, Nov 03 2005 *)

See Mathematica line.
G.f.=zC/(1+z-tz-zC), where C=(1-sqrt(1-4z))/(2z) is the Catalan function. - Emeric Deutsch, Dec 23 2006
With offset 0 (0<=k<=n), T(n,k)=A065600(n,k)+A065600(n+1,k)-A065600(n,k-1). - Philippe Deléham, Apr 01 2007

A132280 Triangle read by rows: T(n,k) is the number of paths in the first quadrant from (0,0) to (n,0), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0), having k H steps (0<=k<=floor(n/2)).

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 9, 6, 1, 21, 16, 3, 51, 45, 12, 1, 127, 126, 40, 4, 323, 357, 135, 20, 1, 835, 1016, 441, 80, 5, 2188, 2907, 1428, 315, 30, 1, 5798, 8350, 4572, 1176, 140, 6, 15511, 24068, 14535, 4284, 630, 42, 1, 41835, 69576, 45925, 15240, 2646, 224, 7

Offset: 0

Emeric Deutsch, Sep 03 2007

Also, T(n,k) is the number of paths in the first quadrant from (0,0) to (n,0), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0), having k peaks (i.e., UD's). Row n contains 1+floor(n/2) terms. T(n,0) = A001006(n) (the Motzkin numbers). Row sums yield A128720. Sum(k*T(n,k), k>=0) = A106053(n).

Shifting every column k of this triangle k steps upwards yields triangle A091869, but with another offset. - Werner Schulte, Feb 03 2017

			Triangle starts:
1;
1;
2,1;
4,2;
9,6,1;
21,16,3;
51,45,12,1;
T(5,2)=3 because we have hHH, HhH and HHh.

Cf. A001006, A106053, A128720.

G:=((1-z-t*z^2-sqrt((1+z-t*z^2)*(1-3*z-t*z^2)))*1/2)/z^2: Gser:=simplify(series(G,z=0,17)): for n from 0 to 13 do P[n]:=sort(coeff(Gser,z,n),n=0..13) end do: for n from 0 to 13 do seq(coeff(P[n],t,j),j=0..floor((1/2)*n)) end do; # yields sequence in triangular form

G.f. = G = G(t,z) satisfies G = 1+zG+tz^2*G+z^2*G^2 (see explicit expression at the Maple program).

Keyword tabf added by Michel Marcus, Apr 09 2013

A355201 Normalized Schur self-convolution expansion coefficients K_{n+1}^n / n giving the coefficients of the Laurent series (compositionally) inverse to f(z) = c_0 z + c_1 + c_2 / z + c_3 / z^2 + ... . Irregular triangle for partition polynomials, with row lengths A000041(n) - 1 except for the first two, which are both of length 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 3, 3, 1, 1, 6, 4, 2, 12, 6, 2, 4, 4, 1, 1, 10, 5, 10, 30, 10, 10, 10, 20, 10, 5, 5, 5, 1, 1, 15, 6, 30, 60, 15, 5, 60, 30, 60, 20, 15, 15, 30, 30, 15, 3, 6, 6, 6, 1, 1, 21, 7, 70, 105, 21, 35, 210, 70, 140, 35, 35, 105, 105, 105, 105, 35, 7, 42, 21, 21, 42, 42, 21, 7, 7, 7, 7, 1

Offset: 0

Tom Copeland, Jun 23 2022

For the formal Laurent series f(z) = a_0 z + a_1 + a_2 / z + a_3 / z^2 + ..., the formal compositional inverse is g(z) = b_0 z + b_1 + b_2 / z + b_3 / z^2 + ..., whose coefficients are partition polynomials whose numerical factors are those of this irregular triangle T(n,k). For the Schur coefficients defined in the formula section, -b_n = K_{n}^{n-1} / (n-1) for n > 1.
Analytic proofs of the relationship between the partition polynomials of the compositional inverse pair of Laurent series and Schur's self-convolution expansion coefficients are given in Schur (beware of a sign error) and in Airault and Ren.
Explicit examples (with a_0=1) of K_{n}^{n-1} up through n=5 are in Airault and Bouali on p. 182.
Various formulas for the b_n in terms of the associahedra (A133437), noncrossing (A134264), reciprocal (A263633), and Faber partition (A263916) polynomials are given in Copeland as well as a derivation of the explicit multi-factorial expression in the formula section and a combinatorial model.

			Triangle begins:
1) 1
2) 1
3) 1
4) 1,  1
5) 1,  1,  2,  1
6) 1,  3,  3,  3,  3,  1
7) 1,  6,  4,  2, 12,  6,  2,  4,  4,  1
8) 1, 10,  5, 10, 30, 10, 10, 10, 20, 10,  5,  5,  5,  1
  ...
The first few partition polynomials, with the monomials in the order of the partitions on p. 831 of Abramowitz and Stegun, are
b0 =    1 / a0
b1 = - a1 / a0
b2 = - a2
b3 = -(a1 a2 + a0 a3)
b4 = -(a1^2 a2 + a0 a2^2 + 2 a0 a1 a3 + a0^2 a4)
b5 = -(a1^3 a2+ 3 a0 a1 a2^2 + 3 a0 a1^2 a_3 + 3 a0^2 a2 a3 + 3 a0^2 a1 a4
      + a0^3 a_5)
b6 = -(a1^4 a2 + 6 a0 a1^2 a2^2 + 4 a0 a1^3 a3 + 2 a0^2 a2^3 + 12 a0^2 a1 a2 a3
      + 6 a0^2 a1^2 a4  + 2 a0^3 a3^2 + 4a0^3 a2 a4 + 4 a0^3 a1 a5 + a0^4 a6)
b7 = -(a1^5 a2 + 10 a_0 a1^3 a2^2 + 5 a0 a1^4 a3 + 10 a0^2 a1 a2^3
      + 30 a0^2 a1^2 a2 a3 + 10 a0^2 a1^3 a4 + 10 a0^3 a2^2 a3 + 10 a0^3 a1 a3^2
      + 20 a0^3 a1 a2 a4 + 10 a0^3 a1^2 a5 + 5 a0^4 a3 a4 + 5 a0^4 a2 a5
      + 5 a0^4 a1 a6 + a0^5 a7)
_____________________

H. Airault and A. Bouali, Differential calculus on the Faber polynomials, Bulletin des Sciences Mathématiques, Volume 130, Issue 3, April-May 2006, pages 179-222.
H. Airault and J. Ren, An algebra of differential operators and generating functions on the set of univalent functions, Bulletin des Sciences Mathématiques, Volume 126, Issue 5, June 2002, pages 343-367.
T. Copeland, One Matrix to Rule Them All: Schur self-Konvolution expansion Koefficients; inversion of Laurent and power series; and associahedra, noncrossing, and reciprocal partition polynomials, 2022.
I. Schur, Identities in the theory of power series, American Journal of Mathematics, Volume 69, No. 1, Jan 1947, pages 14-26.

Cf. A000108, A001263, A091187, A091869, A111785, A133437, A134264, A263633, A263916.

row[0] = row[1] = {1};
row[n_] := With[{s = Expand[Coefficient[Sum[c[k] x^k, {k, 0, n}]^(n-1), x, n] / (n-1)]}, Table[Coefficient[s, Product[c[t], {t, p}]], {p, Reverse[Sort[Sort /@ IntegerPartitions[n, {n-1}, Range[0, n]]]]}]];
Table[row[n], {n, 0, 8}] // Flatten (* Andrey Zabolotskiy, Feb 05 2023 *)

The index notations b(n), b_n, and bn are used interchangeably in this entry for indeterminates.
For n > 1, b_n(a_0,a_1,...,a_n) is a sum of monomials of the form a0^{e0} a1^{e1} a2^{e2} ... an^{en} with e1 * 1 + e2 * 2 + ... + en * n = n. When a_0 is not set to unity, e0 + e1 + ... + en = n - 1. (a1^n is not present.)
From a combinatorial argument in Copeland, the unsigned numerical coefficient of the monomial is given by (n-2)! / [(n - 1 - (e1 + e2 + ... + en))! e1! e2! ... en!].
The partition polynomials are generated by a subset of the Schur self-convolution expansion coefficients as -b_n = K_{n}^{n-1} / (n-1) =(D_{x=0}^n / n!) (a_0 + a_1 x + a_2 x^2 + ... + a_n x^n)^{n-1} / (n-1).
Row sums are the Catalan numbers A000108, ignoring the overall sign, for b_1 onwards.
Reduces to the Narayana triangle A001263 with a_0 = t and all the other indeterminates unity, ignoring the overall sign, for b_2 onwards.
Reduces to A091869 (reversed A091187) with a_1 = t and all the other indeterminates unity, ignoring the overall sign, for b_2 onwards.
b_n(c_1,...,c_n) = - Sum_{k=0}^{n-1} b_k(c_1,...,c_k) N_{n-k}(c_1,...,c_{n-k}) with c_0 = 1 and N_k(c_1,...,c_k) the noncrossing partition polynomials of A134264.
[b] = [R][N], representing the substitution of the noncrossing partition polynomials of A134264 for the indeterminates of the signed reciprocal polynomials of A263633 defined by R_n = 1 / (1 + c_1 x + c_2 x^2 + ...).
Conversely, [R][b] = [N] since the substitution transformation denoted by [R] is an involution, i.e., [R]^2 = [I], the identity substitution.
[b] = [R][A][R], a substitutional conjugation of the set of associahedra partition polynomials of A133147, or A111785, with re-indexing and (1') = 1, e.g., A_0 = 1, A_1 = -c_1, and A_2 = 2 c_1^2 - c_2.
Conversely, [A] = [R][b][R].

Rows 8-9 added by Andrey Zabolotskiy, Feb 05 2023

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.

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A097692 Triangle read by rows: a(n,k) = number of paths of n upsteps U and n downsteps D that contain k UDUs.

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A091187 Triangle read by rows: T(n,k) is the number of ordered trees with n edges and k branches.

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A371408 Number of Dyck paths of semilength n having exactly three (possibly overlapping) occurrences of the consecutive step pattern UDU, where U = (1,1) and D = (1,-1).

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A375253 Expansion of (1 - 2*x + 2*x^2)/(1 - 2*x - 3*x^2)^(7/2).

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A375259 Expansion of (1 - 3*x + 6*x^2 - 4*x^3)/(1 - 2*x - 3*x^2)^(9/2).

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A371411 Number of Dyck paths of semilength 2n having exactly n (possibly overlapping) occurrences of the consecutive step pattern UDU, where U = (1,1) and D = (1,-1).

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A098747 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having exactly k UDU's at low level.

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A375253 Expansion of (1 - 2x + 2x^2)/(1 - 2x - 3x^2)^(7/2).

A375259 Expansion of (1 - 3x + 6x^2 - 4x^3)/(1 - 2x - 3*x^2)^(9/2).