A086810 -id:A086810 - cp's OEIS frontend

A107841 Series reversion of x(1-3x)/(1-x).

1, 2, 10, 62, 430, 3194, 24850, 199910, 1649350, 13879538, 118669210, 1027945934, 9002083870, 79568077034, 708911026210, 6359857112438, 57403123415350, 520895417047010, 4749381474135850, 43489017531266654, 399755692955359630, 3687437532852484442, 34121911117572911410

Offset: 0

Paul Barry, May 24 2005

In general, the series reversion of x(1-r*x)/(1-x) has g.f. (1+x-sqrt(1+2*(1-2*r)*x+x^2))/(2*r) and general term given by a(n)=(1/(n+1))sum{k=0..n, C(n+1,k)C(2n-k,n)(-1)^k*r^(n-k)}; a(n)=(1/(n+1))sum{k=0..n, C(n+1,k+1)C(n+k,k)(-1)^(n-k)*r^k}; a(n)=sum{k=0..n, (1/(k+1))*C(n,k)C(n+k,k)(-1)^(n-k)*r^k}; a(n)=sum{k=0..n, A088617(n,k)*(-1)^(n-k)*r^k}.
The Hankel transform of this sequence is 6^C(n+1,2). - Philippe Deléham, Oct 29 2007
Number of Dyck n-paths with three colors of up (U,a,b) and one color of down (D) avoiding UD. - David Scambler, Jun 24 2013
This sequence is implied in the turbulence solutions of the incompressible Navier-Stokes equations in R^3. a(n) = numbers of realizable vorticity eddies in terms of initial conditions. - Fung Lam, Dec 31 2013
Conjugate sequence to this series is defined by series reversion of x(1+3*x)/(1+x), G.f.: ((x-1)-sqrt(1-10*x+ x^2))/(6*x). Conjugate sequence is the negation of this series except a(0). - Fung Lam, Jan 16 2014
Complete Chebyshev transform is G.f. = 3*F((1-x^2)/(1+x^2)), where F(x) is the g.f. of A107841. Real part of G.f. (= (1 - sqrt(3*x^4-2))/((1+x^2))) generates periodic sequence A056594. In general, for reversion of x*(1-r*x)/(1-x), r>=2, Real part of r*F((1-x^2)/(1+x^2)) (= (1 - sqrt(r*x^4 - r + 1))/(1+x^2)) generates A056594. - Fung Lam, Apr 29 2014
a(n) is the number of small Schröder n-paths with 2 types of up steps (i.e., lattice paths from (0,0) to (2n,0) using steps U1=U2=(1,1), F=(2,0), D=(1,-1), with no F steps on the x-axis). - Yu Hin Au, Dec 07 2019

Fung Lam, Table of n, a(n) for n = 0..1000 [The first 200 terms were computed by Vincenzo Librandi]
J. Abate and W. Whitt, Integer Sequences from Queueing Theory, J. Int. Seq. 13 (2010), 10.5.5, p_n(2).
Yu Hin (Gary) Au, Some Properties and Combinatorial Implications of Weighted Small Schröder Numbers, arXiv:1912.00555 [math.CO], 2019.
Paul Barry, Embedding structures associated with Riordan arrays and moment matrices, arXiv preprint arXiv:1312.0583 [math.CO], 2013.
Z. Chen and H. Pan, Identities involving weighted Catalan-Schroder and Motzkin Paths, arXiv:1608.02448 [math.CO], 2016; eq. (1.13), a=2, b=3.
Samuele Giraudo, Operads from posets and Koszul duality, arXiv preprint arXiv:1504.04529 [math.CO], 2015.
F. Lam, Integral Invariance and Non-linearity Reduction for Proliferating Vorticity Scales in Fluid Dynamics, arXiv:1311.6395 [physics.flu-dyn], 2013-2014.
F. Lam, Vorticity evolution in a rigid pipe of circular cross-section, arXiv preprint arXiv:1505.07723 [physics.flu-dyn], 2015-2019.

Cf. A001003 (r=2), this sequence (r=3), A131763 (r=4), A131765 (r=5), A131846 (r=6), A131926 (r=7), A131869 (r=8), A131927 (r=9).

seq(simplify((-1)^n*hypergeom([-n, n + 1], [2], 3)), n=0..10); # Georg Fischer, Sep 14 2024 (from Peter Luschny's formula in A131763, with last parameter r=3)

CoefficientList[Series[(1+x-Sqrt[1-10*x+x^2])/(6*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)

x='x+O('x^66); Vec(serreverse(x*(1-3*x)/(1-x))) \\ Joerg Arndt, May 15 2013

G.f.: (1+x-sqrt(1-10x+x^2))/(6x).
a(n) = (1/(n+1))sum{k=0..n, C(n+1, k)C(2n-k, n)(-1)^k*3^(n-k)}.
a(n) = (1/(n+1))sum{k=0..n, C(n+1, k+1)C(n+k, k)(-1)^(n-k)*3^k}.
a(n) = sum{k=0..n, (1/(k+1))*C(n, k)C(n+k, k)(-1)^(n-k)*3^k}.
a(n) = sum{k=0..n, A088617(n, k)*(-1)^(n-k)*3^k}.
a(n) = Sum_{k>=0} A086810(n, k)*2^k. - Philippe Deléham, May 26 2005
a(n) = (2/3)*A103210(n) for n>0. - Philippe Deléham, Oct 29 2007
G.f.: 1/(1-2x/(1-3x/(1-2x/(1-3x/(1-2x/(1-3x/(1-2x/(1-3x........ (continued fraction). - Paul Barry, Dec 15 2008
From Paul Barry, May 15 2009: (Start)
G.f.: 1/(1-2x/(1-x-2x/(1-x-2x/(1-x-2x/(1-x-2x/(1-... (continued fraction).
G.f.: 1/(1-2x-6x^2/(1-5x-6x^2/(1-5x-6x^2/(1-5x-6x^2/(1-... (continued fraction). (End)
G.f.: 1/(1+x-3x/(1+x-3x/(1+x-3x/(1+x-3x/(1+x-3x/(1+... (continued fraction). - Paul Barry, Mar 18 2011
D-finite with recurrence: (n+1)*a(n) = 5*(2*n-1)*a(n-1) - (n-2)*a(n-2). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ sqrt(12+5*sqrt(6))*(5+2*sqrt(6))^n/(6*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012
a(n+1) is the coefficient of x^(n+1) in 2*sum{j,1,n}((sum{k,1,n}a(k)x^k)^(j+1)), a(1)=1 with offset by 1. - Fung Lam, Dec 31 2013
The series reversion of x*(1 - r*x)/(1 - x) is D-finite with the general recurrence n*a(n) - (2*r-1)*(2*n-3)*a(n-1) + (n-3)*a(n-2) = 0 and with initial values a(1) = 1, a(2) = r-1, a(3) = (2*r-1)*(r-1). This sequence uses r=3, cf. crossrefs. - Georg Fischer, Sep 14 2024

A126216 Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-1).

Original entry on oeis.org

1, 2, 1, 5, 5, 1, 14, 21, 9, 1, 42, 84, 56, 14, 1, 132, 330, 300, 120, 20, 1, 429, 1287, 1485, 825, 225, 27, 1, 1430, 5005, 7007, 5005, 1925, 385, 35, 1, 4862, 19448, 32032, 28028, 14014, 4004, 616, 44, 1, 16796, 75582, 143208, 148512, 91728, 34398, 7644, 936, 54, 1

Offset: 1

Emeric Deutsch, Dec 20 2006

A Schroeder path of semilength n is a lattice path in the first quadrant, from the origin to the point (2n,0) and consisting of steps U=(1,1), D=(1,-1) and H=(2,0).
Also number of Schroeder paths of semilength n containing exactly k doublerises but no (2,0) steps at level 0 (n >= 1; 0 <= k <= n-1). Also number of doublerise-bicolored Dyck paths (doublerises come in two colors; also called marked Dyck paths) of semilength n and having k doublerises of a given color (n >= 1; 0 <= k <= n-1). Also number of 12312- and 121323-avoiding matchings on [2n] with exactly k crossings.
Essentially the triangle given by [1,1,1,1,1,1,1,1,...] DELTA [0,1,0,1,0,1,0,1,0,1,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 20 2007
Mirror image of triangle A033282. - Philippe Deléham, Oct 20 2007
For relation to Lagrange inversion, or series reversion and the geometry of associahedra, or Stasheff polytopes (and other combinatorial objects), see A133437. - Tom Copeland, Sep 29 2008
First column (k=0) gives the Catalan numbers (A000108). - Alexander Karpov, Jun 10 2018
T(n,k) is the multiplicity of the k-hook representation of the symmetric group in the (n-1)st parking space representation (see Pak and Postnikov, 1995). - Joshua Mundinger, Jul 18 2025

			T(3,1)=5 because we have HUUDD, UUDDH, UUUDDD, UHUDD and UUDHD.
Triangle starts:
   n\k  0      1      2      3      4     5    6   7  8
   1    1;
   2    2,     1;
   3    5,     5;     1;
   4   14,    21,     9,     1;
   5   42,    84,    56,    14,     1;
   6  132,   330,   300,   120,    20,    1;
   7  429,  1287,  1485,   825,   225,   27,   1;
   8 1430,  5005,  7007,  5005,  1925,  385,  35,  1;
   9 4862, 19448, 32032, 28028, 14014, 4004, 616, 44, 1;
  10 ...
Triangle [1,1,1,1,1,1,1,...] DELTA [0,1,0,1,0,1,0,1,...] begins:
   1;
   1,  0;
   2,  1,  0;
   5,  5,  1,  0;
  14, 21,  9,  1,  0;
  42, 84, 56, 14,  1,  0;
  ...

Gheorghe Coserea, Rows n = 1..200, flattened
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
W. Y. C. Chen, T. Mansour and S. H. F. Yan, Matchings avoiding partial patterns, The Electronic Journal of Combinatorics 13, 2006, #R112, Theorem 3.3.
D. Callan, Polygon Dissections and Marked Dyck Paths
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015.
D. Drake, Bijections from Weighted Dyck Paths to Schröder Paths, J. Int. Seq. 13 (2010) #10.9.2.
Rosena R. X. Du, Xiaojie Fan, and Yue Zhao, Enumeration on row-increasing tableaux of shape 2 X n, arXiv:1803.01590 [math.CO], 2018.
Samuele Giraudo, Tree series and pattern avoidance in syntax trees, arXiv:1903.00677 [math.CO], 2019.
S. Mizera, Combinatorics and Topology of Kawai-Lewellen-Tye Relations, arXiv:1706.08527 [hep-th], 2017.
Jean-Christophe Novelli and Jean-Yves Thibon, Duplicial algebras and Lagrange inversion, arXiv preprint arXiv:1209.5959 [math.CO], 2012.
Jean-Christophe Novelli and Jean-Yves Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014. See Fig. 7.
I. Pak and A. Postnikov, Enumeration of trees and one amazing representation of the symmetric group, Proceedings of the 8-th International Conference FPSAC, 1996.

Cf. A000108, A002054, A002055, A002056, A007160, A033280, A033281, A033282.
Cf. A126182.
Cf. A086810, A126120, A243660, A243661.

T:=(n,k)->binomial(n,k)*binomial(2*n-k,n+1)/n: for n from 1 to 11 do seq(T(n,k),k=0..n-1) od; # yields sequence in triangular form

Table[Binomial[n, k] Binomial[2 n - k, n + 1]/n, {n, 10}, {k, 0, n - 1}] // Flatten (* Michael De Vlieger, Jan 09 2016 *)

tabl(nn) = {mP = matrix(nn, nn, n, k, binomial(n-1, k-1)); mN = matrix(nn, nn, n, k, binomial(n-1, k-1) * binomial(n, k-1) / k); mprod = mN*mP; for (n=1, nn, for (k=1, n, print1(mprod[n, k], ", ");); print(););} \\ Michel Marcus, Apr 16 2015

t(n,k) = binomial(n,k)*binomial(2*n-k,n+1)/n;
concat(vector(10, n, vector(n, k, t(n,k-1))))  \\ Gheorghe Coserea, Apr 24 2016

T(n,k) = C(n,k)*C(2*n-k,n+1)/n (0 <= k <= n-1).
G.f.: G(t,z) = (1-2*z-t*z-sqrt(1-4*z-2*t*z+t^2*z^2))/(2*(1+t)*z).
Equals N * P, where N = the Narayana triangle (A001263) and P = Pascal's triangle, as infinite lower triangular matrices. A126182 = P * N. - Gary W. Adamson, Nov 30 2007
G.f.: 1/(1-x-(x+xy)/(1-xy/(1-(x+xy)/(1-xy/(1-(x+xy)/(1-xy/(1-.... (continued fraction). - Paul Barry, Feb 06 2009
Let h(t) = (1-t)^2/(1+(u-1)*(1-t)^2) = 1/(u + 2*t + 3*t^2 + 4*t^3 + ...), then a signed (n-1)-th row polynomial of A126216 is given by u^(2n-1)*(1/n!)*((h(t)*d/dt)^n) t, evaluated at t=0, with initial n=2. The power series expansion of h(t) is related to A181289 (cf. A086810). - Tom Copeland, Oct 09 2011
From Tom Copeland, Oct 10 2011: (Start)
With polynomials
P(0,t) = 0
P(1,t) = 1
P(2,t) = 1
P(3,t) = 2 + t
P(4,t) = 5 + 5 t + t^2
P(5,t) = 14 + 21 t + 9 t^2 + t^3
The o.g.f. A(x,t) = (1+x*t-sqrt((1-x*t)^2-4x))/(2(1+t)), and
B(x,t) = x - x^2/(1-t*x) = x - x^2 - ((t*x)^3 + (t*x)^4 + ...)/t^2 is the compositional inverse in x. [series corrected by Tom Copeland, Dec 10 2019]
Let h(x,t) = 1/(dB/dx) = (1-tx)^2/(1-(t+1)(2x-tx^2)) = 1/(1 - 2x - 3tx^2 + 4t^2x^3 + ...). Then P(n,t) = (1/n!)(h(x,t)*d/dx)^n x, evaluated at x=0, A = exp(x*h(u,t)*d/du) u, evaluated at u=0, and dA/dx = h(A(x,t),t). (End)
From Tom Copeland, Dec 09 2019: (Start)
The polynomials in my 2011 formula entry above evaluate to an aerated, alternating sign sequence of the Catalan numbers A000108 with t = -2. The first few are P(2,-2) = 1, P(3,-2) = 0, P(4,t) = -1, P(5,-2) = 0, P(6,-2) = 2, P(7,-2) = 0, P(8,-2) = -5, P(9,-2) = 0, P(10,-2) = 14.
Generalizing the relations between w = theta and u = phi in Mizera on pp. 32-34, modify the inverse pair above to w = i * B(-i*u,t) = u + i * u^2/(1+i*t*u), where i is the imaginary number, and u = i*A(-i*w,t) = i*(1 - i*w*t - sqrt((1 + i*w*t)^2 + i*4*w))/(2(1+t)). Then the expression for V'(w) in Mizera generalizes to V'(w) = -i*(w - u) and reduces to V'(w) = (1 - sqrt(1-4 w^2))/2 when evaluated at t = -2, which is an o.g.f. for A126120. Cf. also A086810. (End)
Sum_{k = 0..n-1} (-1)^k*T(n,k)*binomial(x + 2*n - k, 2*n - k) = ( (x + 1) * ( Product_{k = 2..n} (x + k)^2 ) * (x + n + 1) )/(n!*(n + 1)!) for n >= 1. Cf. A243660 and A243661. - Peter Bala, Oct 08 2022

A181289 Triangle read by rows: T(n,k) is the number of 2-compositions of n having length k (0 <= k <= n).

Original entry on oeis.org

1, 0, 2, 0, 3, 4, 0, 4, 12, 8, 0, 5, 25, 36, 16, 0, 6, 44, 102, 96, 32, 0, 7, 70, 231, 344, 240, 64, 0, 8, 104, 456, 952, 1040, 576, 128, 0, 9, 147, 819, 2241, 3400, 2928, 1344, 256, 0, 10, 200, 1372, 4712, 9290, 11040, 7840, 3072, 512, 0, 11, 264, 2178, 9108, 22363

Offset: 0

Emeric Deutsch, Oct 12 2010

A 2-composition of n is a nonnegative matrix with two rows, such that each column has at least one nonzero entry and whose entries sum up to n. The length of the 2-composition is the number of columns.
From Tom Copeland, Sep 06 2011: (Start)
R(t,z) = (1-z)^2 / ((1+t)*(1-z)^2-1) = 1/(t - (2*z + 3*z^2 + 4*z^3 + 5*z^4 + ...)) = 1/t + (1/t)^2*2*z + (1/t)^3*(4+3t)*z^2 + (1/t)^4*(8+12*t+4*t^2)*z^3 + ... gives row reversed polynomials of A181289 with G(t,z) = R(1/t,z)/t.
R(t,z) is related to generators for A033282 and A001003 (t=1) and can be umbrally extended to give a partition generator for A133437. (End)
A refined, reverse version of this array is given in A253722. - Tom Copeland, May 02 2015
The infinitesimal generator (infinigen) for the face polynomials of associahedra A086810/A033282, read as decreasing powers, (and for the dual simplicial complex read as increasing powers) can be formed from the row polynomials P(n,t) of this entry. This type of infinigen is presented in A145271 for general sets of binomial Sheffer polynomials. This specific infinigen is presented in analytic form in A086810. Given the column vector of row polynomials V = (P(0,t) = 1, P(1,y) = 2 t, P(2,y) = 3 t + 4 t^2, P(3,y) = 4 t + 12 t^2 + 8 t^3, ...), form the lower triangular matrix M(n,k) = V(n-k,n-k), i.e., diagonally multiply the matrix with all ones on the diagonal and below by the components of V. Form the matrix MD by multiplying A132440^Transpose = A218272 = D (representing derivation of o.g.f.s) by M, i.e., MD = M*D. The non-vanishing component of the first row of (MD)^n * V / (n+1)! is the n-th face polynomial. - Tom Copeland, Dec 11 2015
T is the convolution triangle of the positive integers starting at 2 (see A357368). - Peter Luschny, Oct 19 2022

			Triangle starts:
  1;
  0,  2;
  0,  3,   4;
  0,  4,  12,    8;
  0,  5,  25,   36,   16;
  0,  6,  44,  102,   96,    32;
  0,  7,  70,  231,  344,   240,    64;
  0,  8, 104,  456,  952,  1040,   576,   128;
  0,  9, 147,  819, 2241,  3400,  2928,  1344,   256;
  0, 10, 200, 1372, 4712,  9290, 11040,  7840,  3072,  512;
  0, 11, 264, 2178, 9108, 22363, 34332, 33488, 20224, 6912, 1024;

John Tyler Rascoe, Rows n = 0..100, flattened
G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European J. Combin. 28 (2007), no. 6, 1724-1741.
Y-h. Guo, Some n-Color Compositions, J. Int. Seq. 15 (2012) 12.1.2, eq. (11).

Cf. A003480 (row sums), A181290.
Cf. A253722, A033282, A001003, A133437.
Cf. A086810, A132440, A145271, A218272.
Cf. A000297 (column 3), A006636 (column 4), A006637 (column 5).

T := proc (n, k) if k <= n then sum((-1)^j*2^(k-j)*binomial(k, j)*binomial(n+k-j-1, 2*k-1), j = 0 .. k) else 0 end if end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form
# Uses function PMatrix from A357368.
PMatrix(10, n -> n + 1); # Peter Luschny, Oct 19 2022

Table[Sum[(-1)^j*2^(k - j) Binomial[k, j] Binomial[n + k - j - 1, 2 k - 1], {j, 0, k}], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 11 2015 *)

T_xt(max_row) = {my(N=max_row+1, x='x+O('x^N), h=(1-x)^2/((1-x)^2 - t*x*(2-x))); vector(N, n, Vecrev(polcoeff(h, n-1)))}
T_xt(10) \\ John Tyler Rascoe, Apr 05 2025

T(n,k) = Sum_{j=0..k} (-1)^j*2^(k-j)*binomial(k,j)*binomial(n+k-j-1, 2*k-1) (0 <= k <= n).
G.f.: G(t,x) = (1-x)^2/((1-x)^2 - t*x*(2-x)).
G.f. of column k = x^k*(2-x)^k/(1-x)^{2k} (k>=1) (we have a Riordan array).
Recurrences satisfied by the numbers u_{n,k}=T(n,k) can be found in the Castiglione et al. reference.
Sum_{k=0..n} k*T(n,k) = A181290(n).
T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), T(0,0)=1, T(1,0)=0, T(1,1)=2, T(2,0)=0, T(1,1)=3, T(2,2)=4, T(n,k)=0, if k < 0 or if k > n. - Philippe Deléham, Nov 29 2013

A108263 Triangle read by rows: T(n,k) is the number of short bushes with n edges and k branchnodes (i.e., nodes of outdegree at least two). A short bush is an ordered tree with no nodes of outdegree 1.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 5, 0, 1, 9, 5, 0, 1, 14, 21, 0, 1, 20, 56, 14, 0, 1, 27, 120, 84, 0, 1, 35, 225, 300, 42, 0, 1, 44, 385, 825, 330, 0, 1, 54, 616, 1925, 1485, 132, 0, 1, 65, 936, 4004, 5005, 1287, 0, 1, 77, 1365, 7644, 14014, 7007, 429, 0, 1, 90, 1925, 13650, 34398

Offset: 0

Emeric Deutsch, May 29 2005

Row n has 1+floor(n/2) terms. Row sums are the Riordan numbers (A005043). Column 3 yields A033275; column 4 yields A033276.

Related to the number of certain non-crossing partitions for the root system A_n. Cf. p. 12, Athanasiadis and Savvidou. Diagonals are A033282/A086810. Also see A132081 and A100754.- Tom Copeland, Oct 19 2014

			T(6,3)=5 because the only short bushes with 6 edges and 3 branchnodes are the five full binary trees with 6 edges.
Triangle begins:
1;
0;
0,1;
0,1;
0,1,2;
0,1,5;
0,1,9,5

Indranil Ghosh, Rows 0..100, flattened
C. Athanasiadis and C. Savvidou, The local h-vector of the cluster subdivision of a simplex, arXiv preprint arXiv:1204.0362, 2012
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999) 73-112.
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 10.

Cf. A005043, A033275, A033276.

Columns: A000007, A000012, A000096, A033275, A033276, A033277, A033278, A033279; A005043 (row sums), A033282, A086810

G:=(1+z-sqrt((1-z)^2-4*t*z^2))/2/z/(1+t*z): Gser:=simplify(series(G,z=0,18)): P[0]:=1: for n from 1 to 16 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 16 do seq(coeff(t*P[n],t^k),k=1..1+floor(n/2)) od; # yields sequence in triangular form
A108263 := (n,k) -> binomial(n-k-1,n-2*k)*binomial(n,k)/(n-k+1);
seq(print(seq(A108263(n,k),k=0..ceil((n-1)/2))),n=0..8); # Peter Luschny, Sep 25 2014

T[n_,k_]:=Binomial[n-k-1,n-2k]*Binomial[n,k]/(n-k+1); Flatten[Table[T[n,k],{n,0,11},{k,0,Ceiling[(n-1)/2]}]] (* Indranil Ghosh, Feb 20 2017 *)

G.f. G=G(t, z) satisfies z*(1+t*z)*G^2 - (1+z)*G + 1 = 0.

T(n, k) = A086810(n-k, k). - Philippe Deléham, May 30 2005

A133932 Coefficients of a partition transform for Lagrange inversion of -log(1 - u(.)*t), complementary to A134685 for an e.g.f.

Original entry on oeis.org

1, -1, 3, -2, -15, 20, -6, 105, -210, 40, 90, -24, -945, 2520, -1120, -1260, 420, 504, -120, 10395, -34650, 25200, 18900, -2240, -15120, -9072, 1260, 2688, 3360, -720, -135135, 540540, -554400, -311850, 123200, 415800, 166320, -50400, -56700, -120960, -75600, 18144, 20160, 25920, -5040

Offset: 1

Tom Copeland, Jan 27 2008

Let f(t) = -log(1 - u(.)*t) = Sum_{n>=1} (u_n / n) * t^n.
If u_1 is not equal to 0, then the compositional inverse for f(t) is given by g(t) = Sum_{j>=1} P(n,t) where, with u_n denoted by (n'),
P(1,t) = (1')^(-1) * [ 1 ] * t
P(2,t) = (1')^(-3) * [ -1 (2') ] * t^2 / 2!
P(3,t) = (1')^(-5) * [ 3 (2')^2 - 2 (1')(3') ] * t^3 / 3!
P(4,t) = (1')^(-7) * [ -15 (2')^3 + 20 (1')(2')(3') - 6 (1')^2 (4') ] * t^4 / 4!
P(5,t) = (1')^(-9) * [ 105 (2')^4 - 210 (1') (2')^2 (3') + 40 (1')^2 (3')^2 + 90 (1')^2 (2') (4') - 24 (1')^3 (5') ] * t^5 / 5!
P(6,t) = (1')^(-11) * [ -945 (2')^5 + 2520 (1') (2')^3 (3') - 1120 (1')^2 (2') (3')^2 - 1260 (1')^2 (2')^2 (4') + 420 (1')^3 (3')(4') + 504 (1')^3 (2')(5') - 120 (1')^4 (6') ] * t^6 / 6!
See A134685 for more information.
From Tom Copeland, Sep 28 2016: (Start)
P(7,t) = (1')^(-13) * [ 10395 (2')^6 - 34650 (1')(2')^4(3') + (1')^2 [25200 (2')^2(3')^2 + 18900 (2')^3(4')] - (1')^3 [2240 (3')^3 + 15120 (2')(3')(4') + 9072 (2')^2(5')] + (1')^4 [1260 (4')^2 + 2688 (3')(5') + 3360 (2')(6')] - 720 (1')^5(7')] * t^7 / 7!
P(8,t) = (1')^(-15) * [ -135135 (2')^7 + 540540 (1')(2')^5(3') - (1')^2 [554400 (2')^3(3')^2 + 311850 (2')^4(4')] + (1')^3 [123200 (2')(3')^3 + 415800 (2')^2(3')(4') + 166320 (2')^3(5')] - (1')^4 [50400 (3')^2(4') + 56700 (2')(4')^2 + 120960 (2')(3')(5') + 75600 (2')^2(6')] + (1')^5 [18144 (4')(5') + 20160 (3')(6') + 25920 (2')(7')] - 5040 (1')^6(8')] * t^8 / 8! (End)

			From _Tom Copeland_, Sep 18 2014: (Start)
Let f(x) = log((1-ax)/(1-bx))/(b-a) = -log(1-u.*x) = x + (a+b)x^2/2 + (a^2+ab+b^2)x^3/3 + (a^3+a^2b+ab^2+a^3)x^4/4 + ... . with (u.)^n = u_n = h_(n-1)(a,b) the complete homogeneous polynomials in two indeterminates.
Then the inverse g(x) = (e^(ax)-e^(bx))/(a*e^(ax)-b*e^(bx)) = x - (a+b)x^2/2! + (a^2+4ab+b^2)x^3/3! - (a^3+11a^2b+11ab^2+b^3)x^4/4! + ... , where the bivariate polynomials are the Eulerian polynomials of A008292.
The inversion formula gives, e.g., P(3,x) = 3(u_2)^2 - 2u_3 = 3(h_1)^2 - 2h_2 = 3(a+b)^2 - 2(a^2 + ab + b^2) = a^2 + 4ab + b^2. (End)

T. Copeland, Lagrange a la Lah, 2011.
T. Copeland, Compositional inverse pairs, the Burgers-Hopf equation, and the Stasheff associahedra, 2014.
T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015.
T. Copeland, Formal group laws and binomial Sheffer sequences, 2018.
Jin Wang, Nonlinear Inverse Relations for Bell Polynomials via the Lagrange Inversion Formula, J. Int. Seq., Vol. 22 (2019), Article 19.3.8.

Cf. A145271 (A111999, A007318) = (reduced array, associated g(x)).
Cf. A008292, A086810, A094587, A133314, A134685, A139605, A263633.
Cf. A036039, A133437.

rows[nn_] := {{1}}~Join~With[{s = InverseSeries[t (1 + Sum[u[k] t^k/(k+1), {k, nn}] + O[t]^(nn+1))]}, Table[(n+1)! Coefficient[s, t^(n+1) Product[u[w], {w, p}]], {n, nn}, {p, Reverse[Sort[Sort /@ IntegerPartitions[n]]]}]];
rows[7] // Flatten (* Andrey Zabolotskiy, Mar 08 2024 *)

The bracketed partitions of P(n,t) are of the form (u_1)^e(1) (u_2)^e(2) ... (u_n)^e(n) with coefficients given by (-1)^(n-1+e(1)) * [2*(n-1)-e(1)]! / [ 2^e(2) (e(2))! * 3^e(3) (e(3))! * ... n^e(n) * (e(n))! ].
From Tom Copeland, Sep 06 2011: (Start)
Let h(t) = 1/(df(t)/dt)
  = 1/Ev[u./(1-u.t)]
  = 1/((u_1) + (u_2)*t + (u_3)*t^2 + (u_4)*t^3 + ...),
  where Ev denotes umbral evaluation.
Then for the partition polynomials of A133932,
  n!*P(n,t) = ((t*h(y)*d/dy)^n) y evaluated at y=0,
  and the compositional inverse of f(t) is
  g(t) = exp(t*h(y)*d/dy) y evaluated at y=0.
  Also, dg(t)/dt = h(g(t)). (End)
From Tom Copeland, Oct 20 2011: (Start)
With exp[x* PS(.,t)] = exp[t*g(x)] = exp[x*h(y)d/dy] exp(t*y) eval. at y=0, the raising/creation and lowering/annihilation operators defined by R PS(n,t)=PS(n+1,t) and L PS(n,t)= n*PS(n-1,t) are
R = t*h(d/dt) = t* 1/[(u_1) + (u_2)*d/dt + (u_3)*(d/dt)^2 + ...], and
L = f(d/dt) = (u_1)*d/dt + (u_2)*(d/dt)^2/2 + (u_3)*(d/dt)^3/3 + ....
Then P(n,t) = (t^n/n!) dPS(n,z)/dz  eval. at z=0. (Cf. A139605, A145271, and link therein to Mathemagical Forests for relation to planted trees on p. 13.) (End)
The bracketed partition polynomials of P(n,t) are also given by (d/dx)^(n-1) 1/[u_1 + u_2 * x/2 + u_3 * x^2/3 + ... + u_n * x^(n-1)/n]^n evaluated at x=0. - Tom Copeland, Jul 07 2015
From Tom Copeland, Sep 19 2016: (Start)
Equivalent matrix computation: Multiply the m-th diagonal (with m=1 the index of the main diagonal) of the lower triangular Pascal matrix A007318 by f_m = (m-1)! u_m = (d/dx)^m f(x) evaluated at x=0 to obtain the matrix UP with UP(n,k) = binomial(n,k) f_{n+1-k}, or equivalently, multiply the diagonals of A094587 by u_m. Then P(n,t) = (1, 0, 0, 0,..) [UP^(-1) * S]^(n-1) FC * t^n/n!, where S is the shift matrix A129185, representing differentiation in the basis x^n//n!, and FC is the first column of UP^(-1), the inverse matrix of UP. These results follow from A145271 and A133314.
With u_1 = 1, the first column of UP^(-1) with u_1 = 1 (with initial indices [0,0]) is composed of the row polynomials n! * OP_n(-u_2,...,-u_(n+1)), where OP_n(x[1],...,x[n]) are the row polynomials of A263633 for n > 0 and OP_0 = 1, which are related to those of A133314 as reciprocal o.g.f.s are related to reciprocal e.g.f.s; e.g., UP^(-1)[0,0] = 1, Up^(-1)[1,0] = -u_2, UP^(-1)[2,0] = 2! * (-u_3 + u_2^2) = 2! * OP_2(-u_2,-u_3).
Also, P(n,t) = (1, 0, 0, 0,..) [UP^(-1) * S]^n (0, 1, 0, ..)^T * t^n/n! in agreement with A139605. (End)
From Tom Copeland, Sep 20 2016: (Start)
Let PS(n,u1,u2,...,un) = P(n,t) / (t^n/n!), i.e., the square-bracketed part of the partition polynomials in the expansion for the inverse in the comment section, with u_k = uk.
Also let PS(n,u1=1,u2,...,un) = PB(n,b1,b2,...,bK,...) where each bK represents the partitions of PS, with u1 = 1, that have K components or blocks, e.g., PS(5,1,u2,...,u5) = PB(5,b1,b2,b3,b4) = b1 + b2 + b3 + b4 with b1 = -24 u5, b2 = 90 u2 u4 + 40 u3^2, b3 = -210 u2^2 u3, and b4 = 105 u2^4.
The relation between solutions of the inviscid Burgers's equation and compositional inverse pairs (cf. link and A086810) implies, for n > 2,  PB(n, 0 * b1, 1 * b2, ..., (K-1) * bK, ...) = (1/2) * Sum_{k = 2..n-1} binomial(n+1,k) * PS(n-k+1, u_1=1, u_2, ..., u_(n-k+1)) * PS(k,u_1=1,u_2,...,u_k).
For example, PB(5,0 * b1, 1 * b2, 2 * b3, 3 * b4) = 3 * 105 u2^4 - 2 * 210 u2^2 u3 + 1 * 90 u2 u4 + 1 * 40 u3^2 - 0 * -24 u5 = 315 u2^4 - 420 u2^2 u3 + 90 u2 u4 + 40 u3^2 = (1/2) [2 * 6!/(4!*2!) * PS(2,1,u2) * PS(4,1,u2,...,u4) + 6!/(3!*3!) * PS(3,1,u2,u3)^2] = (1/2) * [ 2 * 6!/(4!*2!) * (-u2) (-15 u2^3 + 20 u2 u3 - 6 u4) + 6!/(3!*3!) * (3 u2^2 - 2 u3)^2].
Also, PB(n,0*b1,1*b2,...,(K-1)*bK,...) =  d/dt t^(n-2)*PS(n,u1=1/t,u2,...,un)|{t=1} = d/dt (1/t)*PS(n,u1=1,t*u2,...,t*un)|{t=1}.
(End)
A recursion relation for computing each partition polynomial of this entry from the lower order polynomials and the coefficients of the refined Stirling polynomials of the first kind A036039 is presented in the blog entry "Formal group laws and binomial Sheffer sequences." - Tom Copeland, Feb 06 2018

Terms ordered according to the reversed Abramowitz-Stegun ordering of partitions (with every k' replaced by (k-1)') by Andrey Zabolotskiy, Mar 08 2024

A131763 Series reversion of x(1-4x)/(1-x) is x*A(x) where A(x) is the generating function.

Original entry on oeis.org

1, 3, 21, 183, 1785, 18651, 204141, 2310447, 26819121, 317530227, 3819724293, 46553474919, 573608632233, 7133530172619, 89423593269213, 1128765846337887, 14334721079385441, 183021615646831587, 2347944226115977461, 30250309354902101271, 391241497991342192985

Offset: 0

Philippe Deléham, Oct 29 2007, Nov 06 2007

nonn

The Hankel transform of this sequence is 12^C(n+1,2).
Number of Dyck n-paths with two colors of up (U,u) and two colors of down (D,d) avoiding UD. - David Scambler, Jun 24 2013
Number of small Schröder n-paths with 3 types of up steps (i.e., lattice paths from (0,0) to (2n,0) using steps U1=U2=U3=(1,1), F=(2,0), D=(1,-1), with no F steps on the x-axis). - Yu Hin Au, Dec 05 2019

			G.f. = 1 + 3*x + 21*x^2 + 183*x^3 + 1785*x^4 + 18651*x^5 + ... - _Michael Somos_, Jul 27 2022

J. Abate and W. Whitt, Integer Sequences from Queueing Theory, J. Int. Seq. 13 (2010), 10.5.5, p_n(3).
Yu Hin (Gary) Au, Some Properties and Combinatorial Implications of Weighted Small Schröder Numbers, arXiv:1912.00555 [math.CO], 2019.
Paul Barry, Embedding structures associated with Riordan arrays and moment matrices, arXiv:1312.0583 [math.CO], 2013.
Z. Chen and H. Pan, Identities involving weighted Catalan-Schroder and Motzkin Paths, arXiv:1608.02448 [math.CO], (2016), eq. (1.13), a=3, b=4.

Cf. A086810, A103211.

Cf. for series reversion of x*(1-r*x)/(1-x): A001003 (r=2), A107841 (r=3), this sequence (r=4), A131765 (r=5), A131846 (r=6), A131926 (r=7), A131869 (r=8), A131927 (r=9).

Rest[CoefficientList[InverseSeries[Series[x*(1-4*x)/(1-x), {x, 0, 20}], x],x]] (* Vaclav Kotesovec, Mar 30 2015 *)
Table[(-1)^n Hypergeometric2F1[-n, n + 1, 2, 4], {n, 0, 20}] (* Peter Luschny, Jan 08 2018 *)
a[ n_] := SeriesCoefficient[(1 + x - Sqrt[1 - 14*x + x^2])/(8*x), {x, 0, n}]; (* Michael Somos, Jul 27 2022 *)
a[ n_] := (-1)^n * Hypergeometric2F1[ -n, n+1, 2, 4]; (* Michael Somos, Mar 15 2024 *)

Vec(serreverse(x*(1-4*x)/(1-x)+ O(x^30))) \\ Michel Marcus, Mar 30 2015

{a(n) = if(n<0, 0, n++; polcoeff(serreverse(x*(1-4*x)/(1-x) + x*O(x^n)), n))}; /* Michael Somos, Jul 27 2022 */

{a(n) = if(n<0, -a(-1-n), polcoeff(2/(1 + x + sqrt(1 - 14*x + x^2 + x*O(x^n))), n))}; /* Michael Somos, Mar 15 2024 */

a(n) = Sum_{0<=k<=n} A086810(n,k)*3^k.
a(n) = (3/4)*A103211(n) for n>0.
a(n) = -a(n-1)+4*Sum_{i=0..n-1} a(i)*a(n-i-1), a(0)=1. - Vladimir Kruchinin, Mar 30 2015
D-finite with recurrence: (n+1)*a(n) +7*(-2*n+1)*a(n-1) +(n-2)*a(n-2)=0. - R. J. Mathar, Aug 16 2015
a(n) = (-1)^n*hypergeom([-n, n + 1], [2], 4). - Peter Luschny, Jan 08 2018
G.f.: (1 + x - sqrt(1 - 14*x + x^2))/(8*x). - Michael Somos, Jul 27 2022
From Michael Somos, Mar 15 2024: (Start)
Given g.f. A(x) and y = 2*x*A(-x^2), then y-1/y = (x-1/x)/2.
If a(n) := -a(-1-n) for n<0, then 0 = a(n)*(+a(n+1) -35*a(n+2) +4*a(n+3)) +a(n+1)*(+7*a(n+1) +194*a(n+2) -35*a(n+3)) +a(n+2)*(+7*a(n+2) +a(n+3)) for all n in Z. (End)

a(17) corrected by Mark van Hoeij, Jul 01 2010

A131765 Series reversion of x*(1-5x)/(1-x) .

Original entry on oeis.org

1, 4, 36, 404, 5076, 68324, 963396, 14046964, 210062196, 3204118724, 49656709476, 779690085204, 12376867734036, 198301332087204, 3202580085625476, 52080967814444724, 852103170531254196, 14016301507253656964

Offset: 0

Philippe Deléham, Oct 29 2007

nonn

The Hankel transform of this sequence is 20^C(n+1,2).

a(n) is the number of small Schröder n-paths with 4 types of up steps (i.e., lattice paths from (0,0) to (2n,0) using steps U1=U2=U3=U4=(1,1), F=(2,0), D=(1,-1), with no F steps on the x-axis). - Yu Hin Au, Dec 05 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Yu Hin (Gary) Au, Some Properties and Combinatorial Implications of Weighted Small Schröder Numbers, arXiv preprint, arXiv:1912.00555 [math.CO], 2019.

Cf. A000108, A086810.

Table[Sum[Binomial[n+k,2*k]*Binomial[2*k,k]/(k+1)*(-1)^(n-k)*5^k,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 20 2012 *)
a[n_] := Sum[(-1)^k Binomial[n, k] Hypergeometric2F1[k - n, n + 1, k + 2, -4], {k, 0, n}]; Table[a[n], {n, 0, 17}] (* Peter Luschny, Jan 08 2018 *)

Vec(serreverse(x*(1-5*x)/(1-x) + O(x^30))) \\ Michel Marcus, Jan 08 2018

a(n) = Sum_{k=0..n} A086810(n,k)*4^k.
From Paul Barry, Sep 08 2009: (Start)
a(n) = Sum_{k=0..n} C(n+k,2*k)*A000108(k)*(-1)^(n-k)*5^k;
a(n) = Sum_{k=0..n} C(n+k,2*k)*A000108(k)*(4^(k+1)+(-1)^k)/5. (End)
Recurrence: (n+1)*a(n) = 9*(2*n-1)*a(n-1) - (n-2)*a(n-2) . - Vaclav Kotesovec, Oct 20 2012
a(n) ~ sqrt(40+18*sqrt(5))*(9+4*sqrt(5))^n/(10*sqrt(Pi)*n^(3/2)) . - Vaclav Kotesovec, Oct 20 2012. Equivalently, a(n) ~ phi^(6*n + 3) / (sqrt(2) * 5^(3/4) * sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 07 2021
a(n) = Sum_{k=0..n} (-1)^k*binomial(n, k)*hypergeom([k - n, n + 1], [k + 2], -4). - Peter Luschny, Jan 08 2018

Extra terms added by Paul Barry, Sep 08 2009

A133336 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,1,1,1,1,1,1,...] DELTA [0,1,0,1,0,1,0,1,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 5, 5, 1, 0, 14, 21, 9, 1, 0, 42, 84, 56, 14, 1, 0, 132, 330, 300, 120, 20, 1, 0, 429, 1287, 1485, 825, 225, 27, 1, 0, 1430, 5005, 7007, 5005, 1925, 385, 35, 1, 0, 4862, 19448, 32032, 28028, 14014, 4004, 616, 44, 1, 0, 16796, 75582, 143208, 148512, 91728, 34398, 7644, 936, 54, 1, 0

Offset: 0

Philippe Deléham, Oct 19 2007

Mirror image of triangle A086810; another version of A126216.
Equals A131198*A007318 as infinite lower triangular matrices. - Philippe Deléham, Oct 23 2007
Diagonal sums: A119370. - Philippe Deléham, Nov 09 2009

			Triangle begins:
    1;
    1,    0;
    2,    1,    0;
    5,    5,    1,   0;
   14,   21,    9,   1,   0;
   42,   84,   56,  14,   1,  0;
  132,  330,  300, 120,  20,  1, 0;
  429, 1287, 1485, 825, 225, 27, 1, 0;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
W. Y. C. Chen, T. Mansour and S. H. F. Yan, Matchings avoiding partial patterns, The Electronic Journal of Combinatorics 13, 2006, #R112, Theorem 3.3.

Cf. A000108, A002054, A002055, A002056, A007160, A033280, A033281, A033282.

[[Binomial(n-1,k)*Binomial(2*n-k,n)/(n+1): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Feb 05 2018

Table[Binomial[n-1,k]*Binomial[2*n-k,n]/(n+1), {n,0,10}, {k,0,n}] // Flatten (* G. C. Greubel, Feb 05 2018 *)

for(n=0,10, for(k=0,n, print1(binomial(n-1,k)*binomial(2*n-k,n)/(n+1), ", "))) \\ G. C. Greubel, Feb 05 2018

Sum_{k=0..n} T(n,k)*x^k = A000108(n), A001003(n), A007564(n), A059231(n), A078009(n), A078018(n), A081178(n), A082147(n), A082181(n), A082148(n), A082173(n) for x = 0,1,2,3,4,5,6,7,8,9,10 respectively.
Sum_{k=0..n} T(n,k)*x^(n-k) = A000007(n), A001003(n), A107841(n), A131763(n), A131765(n), A131846(n), A131926(n), A131869(n), A131927(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively. - Philippe Deléham, Nov 05 2007
Sum_{k=0..n} T(n,k)*(-2)^k*5^(n-k) = A152601(n). - Philippe Deléham, Dec 10 2008
T(n,k) = binomial(n-1,k)*binomial(2n-k,n)/(n+1), k <= n. - Philippe Deléham, Nov 02 2009

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A107841 Series reversion of x*(1-3*x)/(1-x).

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A126216 Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-1).

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A181289 Triangle read by rows: T(n,k) is the number of 2-compositions of n having length k (0 <= k <= n).

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A108263 Triangle read by rows: T(n,k) is the number of short bushes with n edges and k branchnodes (i.e., nodes of outdegree at least two). A short bush is an ordered tree with no nodes of outdegree 1.

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A133932 Coefficients of a partition transform for Lagrange inversion of -log(1 - u(.)*t), complementary to A134685 for an e.g.f.

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A131763 Series reversion of x*(1-4*x)/(1-x) is x*A(x) where A(x) is the generating function.

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A131765 Series reversion of x*(1-5x)/(1-x) .

A107841 Series reversion of x(1-3x)/(1-x).

A131763 Series reversion of x(1-4x)/(1-x) is x*A(x) where A(x) is the generating function.

A131846 Expansion of series reversion of x(1-6x)/(1-x).

A131869 Expansion of series reversion of x(1-8x)/(1-x).