A052701 -id:A052701 - cp's OEIS frontend

A192404 G.f. satisfies: A(x,y) = 1 + Sum_{n>=1} x^nyA(x,y)^n/(1 - yA(x,y)^(2n)), where A(x,y) = 1 + Sum_{n>=1,k>=1} T(n,k)x^ny^k; here the coefficients T(n,k) form a square array and are read by antidiagonals.

1, 1, 1, 1, 2, 1, 1, 4, 5, 1, 1, 7, 14, 10, 1, 1, 11, 31, 38, 17, 1, 1, 16, 61, 114, 91, 26, 1, 1, 22, 111, 291, 357, 196, 37, 1, 1, 29, 190, 656, 1131, 971, 384, 50, 1, 1, 37, 309, 1345, 3092, 3771, 2367, 694, 65, 1, 1, 46, 481, 2563, 7575, 12393, 11150, 5286, 1173, 82, 1

Offset: 1

Paul D. Hanna, Jun 30 2011

Related q-series identity:

Sum_{n>=1} x^n*y*q^n/(1-y*q^(2*n)) = Sum_{n>=1} y^n*x*q^(2*n-1)/(1-x*q^(2*n-1)); here q=A(x,y).

			Let A = g.f. A(x,y), then A satisfies the following relations:
A = 1 + x*y*A/(1-y*A^2) + x^2*y*A^2/(1-y*A^4) + x^3*y*A^3/(1-y*A^6) +...
A = 1 + y*x*A/(1-x*A) + y^2*x*A^3/(1-x*A^3) + y^3*x*A^5/(1-x*A^5) +...
The square array of coefficients in A(x,y) begins:
[1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,...];
[0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...];
[0,1,2,5,10,17,26,37,50,65,82,101,122,145,170,197,226,257,...];
[0,1,4,14,38,91,196,384,694,1173,1876,2866,4214,5999,8308,...];
[0,1,7,31,114,357,971,2367,5286,10969,21367,39391,69202,...];
[0,1,11,61,291,1131,3771,11150,29828,73329,167767,360791,...];
[0,1,16,111,656,3092,12393,43464,136434,390259,1031194,...];
[0,1,22,190,1345,7575,35839,146712,533050,1751371,5278494,...];
[0,1,29,309,2563,17011,93599,441925,1838094,6865479,...];
[0,1,37,481,4609,35563,224947,1212807,5721008,24088842,...];
[0,1,46,721,7906,70021,504448,3078603,16340045,77009425,...]; ...
in which the zeroth row and column are omitted from this sequence.
EXPLICIT EXPANSIONS of the generating function are as follows.
SUMMATION ALONG ROWS gives:
A(x,y) = 1 + x*y/(1-y) + x^2*(y - y^2 + 2*y^3)/(1-y)^3 + x^3*(y - y^2 + 4*y^3 - 2*y^4 + 6*y^5)/(1-y)^5 + x^4*(y + 3*y^3 + 9*y^4 + 5*y^6 + 22*y^7)/(1-y)^7 + x^5*(y + 2*y^2 - 2*y^3 + 54*y^4 - 90*y^5 + 204*y^6 - 133*y^7 + 98*y^8 + 90*y^9)/(1-y)^9 +...
in which the denominator polynomials are the odd powers of (1-y).
The coefficients of y in the above numerator polynomials begin:
[1];
[1,-1,2];
[1,-1,4,-2,6];
[1,0,3,9,0,5,22];
[1,2,-2,54,-90,204,-133,98,90];
[1,5,-10,150,-329,964,-1339,2025,-1385,868,394];
[1,9,-18,305,-667,2377,-3763,7967,-10012,14378,-10323,6388,1806];
[1,14,-21,518,-819,3536,-2367,6387,-1660,20583,-35708,77417,-64888,43361,8558];...
in which the row sums form A052701 (related to Catalan numbers), and the rightmost border forms the large Schroeder numbers (A006318).
SUMMATION ALONG COLUMNS gives:
A(x,y) = 1 + y*x/(1-x) + y^2*(x - x^2 + x^3)/(1-x)^3 + y^3*(x - x^3 + x^4 + x^5)/(1-x)^5 + y^4*(x + 3*x^2 - 11*x^3 + 23*x^4 - 24*x^5 + 12*x^6 + x^7)/(1-x)^7 + y^5*(x + 8*x^2 - 26*x^3 + 66*x^4 - 108*x^5 + 137*x^6 - 117*x^7 + 52*x^8 + x^9)/(1-x)^9 ...
in which the denominator polynomials are the odd powers of (1-x).
The coefficients of x in the above numerator polynomials begin:
[1];
[1,-1,1];
[1,0,-1,1,1];
[1,3,-11,23,-24,12,1];
[1,8,-26,66,-108,137,-117,52,1];
[1,15,-35,80,-90,95,-164,330,-377,186,1];
[1,24,-19,-25,464,-1516,3075,-4066,3274,-928,-778,625,1];
[1,35,49,-329,2023,-6479,15515,-28703,41895,-46744,36552,-15870,429,2054,1];...
in which the row sums form the Catalan numbers (A000108).

Cf. A192405 (antidiagonal sums), A192406 (main diagonal), A192407.

{T(n,k)=local(A=1+x*y);for(i=1,n+k,A=1+sum(m=1,n+k,x^m*y*A^m/(1-y*A^(2*m)+x*O(x^n)+y*O(y^k))));polcoeff(polcoeff(A,n,x),k,y)}

{T(n,k)=local(A=1+x*y);for(i=1,n+k,A=1+sum(m=1,n+k,y^m*x*A^(2*m-1)/(1-x*A^(2*m-1)+x*O(x^n)+y*O(y^k))));polcoeff(polcoeff(A,k,y),n,x)}
/* Print the coefficients as a square array: */
{for(n=1,12,for(k=1,18-n,print1(T(n,k),","));print(""))}
/* Print the array in flattened format: */
{for(n=1,12,for(k=1,n,print1(T(n-k+1,k),","));)}

G.f. satisfies: A(x,y) = 1 + Sum_{n>=1} y^n*x*A(x,y)^(2*n-1)/(1 - x*A(x,y)^(2*n-1)).

A234939 Coefficients of Hilbert series for suboperad of bicolored noncrossing configurations generated by a triangle with colored base and at least one more colored edge and a triangle with one colored non-base edge.

Original entry on oeis.org

1, 2, 8, 38, 200, 1124, 6608, 40142, 249992, 1587548, 10241264, 66926204, 442120016, 2947660616, 19808372384, 134030802782, 912385334792, 6244056445868, 42935538999728, 296493196682036, 2055313327353200, 14297177397185912, 99769106353379168, 698228176760193068

Offset: 1

N. J. A. Sloane, Jan 04 2014

nonn

Frédéric Chapoton and Samuele Giraudo, Enveloping operads and bicoloured noncrossing configurations, arXiv preprint arXiv:1310.4521 [math.CO], 2013-2014.

Cf. A234938, A052701, A007863, A006013, A006318, A007564.

a(n) = 2 * A007564(n-1) for n > 1 [from Chapoton & Giraudo, Proposition 3.8]. - Andrey Zabolotskiy, Feb 01 2025

Terms a(9) onwards added and name clarified by Andrey Zabolotskiy, Feb 02 2025

A090375 Number of unrooted Eulerian maps with bicolored faces which are self-isomorphic under reversing the colors.

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 40, 93, 224, 538, 1344, 3352, 8448, 21573, 54912, 143037, 366080, 968083, 2489344, 6664856, 17199104, 46515759, 120393728, 328382874, 852017152, 2340706462, 6085836800, 16822999572, 43818024960, 121777594508, 317680680960, 887053276477

Offset: 1

Valery A. Liskovets, Dec 01 2003

M. Deryagina, On the enumeration of hypermaps which are self-equivalent with respect to reversing the colors of vertices, Preprint 2016.
V. Liskovets, Some easily derivable integer sequences, J. Integer Seq., v.3 (2000), Article 00.2.2.
V. A. Liskovets, Enumerative formulas for unrooted planar maps: a pattern, Electron. J. Combin., 11:1 (2004), R88.

Cf. A052701, A069727, A090371.

A069727[n_] := (1/(2n)) (3*2^(n - 1) Binomial[2 n, n]/((n + 1) (n + 2)) + Sum[EulerPhi[n/k] d[n/k] 2^(k - 2) Binomial[2 k, k], {k, Most[Divisors[n]]}]) + q[n]; A069727[0] = 1;
q[n_?EvenQ] := 2^((n - 4)/2) Binomial[n, n/2]/(n + 2); q[n_?OddQ] := 2^((n - 1)/2) Binomial[(n - 1), (n - 1)/2]/(n + 1);
d[n_] := 4 - Mod[n, 2];
h0[n_] := 3*2^(n - 1) Binomial[2n, n]/((n + 1)(n + 2));
A090371[n_] := (h0[n] + DivisorSum[n, If[# > 1, EulerPhi[#]*Binomial[n/# + 2, 2] h0[n/#], 0] &])/n;
a[n_] := 2 A069727[n] - A090371[n];
Array[a, 32] (* Jean-François Alcover, Aug 28 2019 *)

h0(n) = 3*2^(n-1)*binomial(2*n, n)/((n+1)*(n+2));
a090371(n) = (h0(n) + sumdiv(n, d, (d>1)*eulerphi(d)*binomial(n/d+2, 2)*h0(n/d)))/n;
d(n) = if (n%2, 3, 4);
q(n) = if (n%2, 2^((n-1)/2)*binomial(n-1, (n-1)/2)/(n+1), 2^((n-4)/2)*binomial(n, n/2)/(n+2));
a069727(n) = if (n==0, 1, q(n) + (3*2^(n-1)*binomial(2*n, n)/((n+1)*(n+2)) + sumdiv(n, k, (k!=n)*eulerphi(n/k)*d(n/k)*2^(k-2)*binomial(2*k, k)))/(2*n));
a(n) = 2*a069727(n) - a090371(n); \\ Michel Marcus, Dec 11 2014

a(n) = 2*A069727(n) - A090371(n).

a(2k+1) = 2^k*Catalan(k) = A052701(k+1).

More terms from Michel Marcus, Dec 11 2014

A101477 Square array T(n,k), read by antidiagonals: number of labeled trees, with increments of labels along edges constrained to +-1, with n nodes that have no label greater than k.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 2, 7, 12, 1, 2, 8, 31, 56, 1, 2, 8, 39, 156, 288, 1, 2, 8, 40, 211, 851, 1584, 1, 2, 8, 40, 223, 1219, 4909, 9152, 1, 2, 8, 40, 224, 1327, 7371, 29506, 54912, 1, 2, 8, 40, 224, 1343, 8250, 46099, 183043, 339456, 1, 2, 8, 40, 224, 1344, 8427, 52938, 295915, 1164387, 2149888

Offset: 0

Ralf Stephan, Jan 21 2005

			1, 1, 3, 12,  56,  288, 1584,  9152,  54912,  339456, ...
1, 2, 7, 31, 156,  851, 4909, 29506, 183043, 1164387, ...
1, 2, 8, 39, 211, 1219, 7371, 46099, 295915, 1939395, ...
1, 2, 8, 40, 223, 1327, 8250, 52938, 347941, 2330532, ...
1, 2, 8, 40, 224, 1343, 8427, 54625, 362833, 2456261, ...
1, 2, 8, 40, 224, 1344, 8447, 54887, 365688, 2484384, ...
1, 2, 8, 40, 224, 1344, 8448, 54911, 366051, 2488831, ...
1, 2, 8, 40, 224, 1344, 8448, 54912, 366079, 2489311, ...
1, 2, 8, 40, 224, 1344, 8448, 54912, 366080, 2489343, ...
1, 2, 8, 40, 224, 1344, 8448, 54912, 366080, 2489344, ...

M. Bousquet-Mélou, Limit laws for embedded trees, arXiv:math/0501266 [math.CO], 2005.

Rows converge to A052701. First row is A000257.

Cf. A052701, A101478.

nmax = 11;
b[x_] = Sum[2^(n - 1)*(2*n - 2)!/(n - 1)!/n! x^n, {n, 1, nmax}];
c[x_] = 0; Do[c[x_] = x*(1 + c[x])^4/(1 + c[x]^2) + O[x]^nmax, {nmax}];
a[n_, t_] := a[n, t] = b[t]*(1 - c[t]^(n + 1))*(1 - c[t]^(n + 5))/((1 - c[t]^(n + 2))*(1 - c[t]^(n + 4)));
T[n_, k_] := SeriesCoefficient[a[n, t], {t, 0, k}];
Table[T[n - k, k], {n, 1, nmax}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 25 2018 *)

G.f. of k-th row: A(t)=B(t)*(1-C(t)^(k+1))*(1-C(t)^(k+5))/[(1-C(t)^(k+2))*(1-C(t)^(k+4))], with tB(t) the g.f. of A052701 and C(t) the g.f. of A101478.

A101596 G.f.: c(2*x)^4, where c(x) is the g.f. of A000108.

Original entry on oeis.org

1, 8, 56, 384, 2640, 18304, 128128, 905216, 6449664, 46305280, 334721024, 2434334720, 17801072640, 130809692160, 965500108800, 7154863964160, 53214300733440, 397094950010880, 2972195534929920, 22308469918924800

Offset: 0

Paul Barry, Dec 08 2004

a(n) is also the number of paths in a binary tree of length 2n+3 between two vertices that are 3 steps apart. - David Koslicki, (koslicki(AT)math.psu.edu), Nov 02 2010

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

Cf. A085687, A003645, A052701.

CoefficientList[Series[(1-Sqrt[1-8z]+4z(-2+Sqrt[1-8z]+2z))/(32z^4), {z, 0, 20}],z] (* Benedict W. J. Irwin, Jul 12 2016 *)

x='x+O('x^50); Vec((1-sqrt(1-8*x) + 4*x*(2*x-2+ sqrt(1-8*x)) )/(32*x^4)) \\ G. C. Greubel, May 24 2017

a(n) = ((8*n+12)/(3*n+12))*((3*n+3)/(n+3))*2^n*C(n+1), where C(n) and the Catalan numbers of A000108.
Conjecture: (n+4)*a(n)-4*(3n+7)*a(n-1)+16*(2n+1)*a(n-2)=0. - R. J. Mathar, Dec 13 2011
From Benedict W. J. Irwin, Jul 12 2016: (Start)
G.f.: (1-sqrt(1-8*x)+4*x*(2*x-2+sqrt(1-8*x)))/(32*x^4).
E.g.f: E^(4*x)*(2*x*(4*x-3)*BesselI(0,4*x) + (3-4*x+ 8*x^2)* BesselI(1, 4*x))/(4*x^3). (End)
a(n) ~ 2^(3*n+5)*n^(-3/2)/sqrt(Pi). - Ilya Gutkovskiy, Jul 12 2016

A114608 Triangle read by rows: T(n,k) is the number of bicolored Dyck paths of semilength n and having k peaks of the form ud (0 <= k <= n). A bicolored Dyck path is a Dyck path in which each up-step is of two kinds: u and U.

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 11, 19, 9, 1, 45, 96, 66, 16, 1, 197, 501, 450, 170, 25, 1, 903, 2668, 2955, 1520, 365, 36, 1, 4279, 14407, 18963, 12355, 4165, 693, 49, 1, 20793, 78592, 119812, 94528, 41230, 9856, 1204, 64, 1, 103049, 432073, 748548, 693588, 372078, 117054

Offset: 0

Emeric Deutsch, Dec 15 2005

Row sums yield A052701. Column 0 yields the little Schroeder numbers (A001003). Sum_{k=0..n} k*T(n,k) = A069720(n).

Triangle T(n,k), 0 <= k <= n, read by rows; given by [1, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 23 2005

			T(3,2)=9 because we have (ud)(ud)Ud, (ud)Ud(ud), Ud(ud)(ud), (ud)u(ud)d,
(ud)U(ud)d, u(ud)d(ud), U(ud)d(ud), u(ud)(ud)d and U(ud)(ud)d (the ud peaks are shown between parentheses).
Triangle starts:
   1;
   1,  1;
   3,  4,  1;
  11, 19,  9,  1;
  45, 96, 66, 16,  1;

Paul Barry, Generalized Eulerian Triangles and Some Special Production Matrices, arXiv:1803.10297 [math.CO], 2018.

Cf. A001003, A052701, A069720.

T:=proc(n,k) if k<=n-1 then (1/n)*binomial(n,k)*sum(2^j*binomial(n,j+1)*binomial(n-k,j),j=0..n-k) elif k=n then 1 else 0 fi end: for n from 0 to 10 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

T[n_, k_] := If[k <= n-1, (1/n)*Binomial[n, k]*Sum[2^j*Binomial[n, j+1]* Binomial[n-k, j], {j, 0, n-k}], If[k == n, 1, 0]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 11 2018, from Maple *)

T(n,k) = (1/n)*binomial(n,k)*Sum_{j=0..n-k} 2^j*binomial(n, j+1)*binomial(n-k, j) (k <= n-1); T(n, n)=1.

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

A117505 Triangle of coefficients for polynomials used for the column g.f.s of triangle A116880, called CM(1,2).

Original entry on oeis.org

1, 2, 1, 2, 4, 3, 2, 4, 16, 13, 2, 4, 16, 80, 67, 2, 4, 16, 80, 448, 381, 2, 4, 16, 80, 448, 2688, 2307, 2, 4, 16, 80, 448, 2688, 16896, 14589, 2, 4, 16, 80, 448, 2688, 16896, 109824, 95235, 2, 4, 16, 80, 448, 2688

Offset: 0

Wolfdieter Lang, Apr 13 2006

The g.f. G(m,x) for column m=1,2,... of triangle A116880=CM(1,2) is x*(-sum(a(m,k)*x^(k-1),k=1..m) + sum(a(m,k)*x^k,k=0..m)*2*c(2*x))/(1+x), with the o.g.f. c(x) of A000108 (Catalan numbers).

			m=3: G(3,x)= x*(-(4+16*x+13*x^2) +
(2+4*x+16*x^2+13*x^3)*2*c(2*x))/(1+x).

W. Lang: First 10 rows.

a(m,m)= A064062(m) =:C(2;m), m>=0 and a(m,k)=2*A052701(k) = C(k)*2^(k+1), for k=1,...,m-1 and C(k):=A000108(k) (Catalan).

A234938 Coefficients of Hilbert series for the suboperad of bicolored noncrossing configurations generated by a fully colored triangle and a fully uncolored triangle.

Original entry on oeis.org

1, 2, 8, 40, 216, 1246, 7516, 46838, 299200, 1948804, 12893780, 86415940, 585461380, 4003022222, 27587072156, 191426864328, 1336331235624, 9378578814890, 66133103587412, 468323884345060, 3329180643569660, 23748479467116032, 169944228206075568, 1219639212041064130

Offset: 1

N. J. A. Sloane, Jan 04 2014

nonn

Frédéric Chapoton and Samuele Giraudo, Enveloping operads and bicoloured noncrossing configurations, arXiv preprint arXiv:1310.4521 [math.CO], 2013-2014.

Cf. A234939, A052701, A007863, A006013, A006318.

Rest@CoefficientList[Root[Function[{f}, 4t-2t^2-t^3+t^4 + (-4+4t-t^2+2t^3)f + (6+t)f^2 + (1-2t)f^3 - f^4], 2] + O[t]^25, t] (* Andrey Zabolotskiy, Feb 02 2025 *)

G.f. A(t) satisfies 4t-2t^2-t^3+t^4 + (-4+4t-t^2+2t^3)*A(t) + (6+t)*A(t)^2 + (1-2t)*A(t)^3 - A(t)^4 = 0 [Chapoton & Giraudo, Proposition 3.5]. - Andrey Zabolotskiy, Feb 02 2025

Terms a(9) onwards added and name clarified by Andrey Zabolotskiy, Feb 02 2025

A214582 Riordan array (1/(1-x-x^2), x(1+2x)).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 3, 4, 5, 1, 5, 7, 10, 7, 1, 8, 11, 15, 20, 9, 1, 13, 18, 25, 35, 34, 11, 1, 21, 29, 40, 55, 75, 52, 13, 1, 34, 47, 65, 90, 125, 143, 74, 15, 1, 55, 76, 105, 145, 200, 275, 247, 100, 17, 1

Offset: 0

Philippe Deléham, Mar 06 2013

First column is A000045 (Fibonacci numbers) starting with 1.

Second column is A000032 (Lucas numbers) starting with 1.

			Triangle begins
1
1, 1
2, 3, 1
3, 4, 5, 1
5, 7, 10, 7, 1
8, 11, 15, 20, 9, 1
13, 18, 25, 35, 34, 11, 1
21, 29, 40, 55, 75, 52, 13, 1
34, 47, 65, 90, 125, 143, 74, 15, 1
55, 76, 105, 145, 200, 275, 247, 100, 17, 1
...
Production array begins
1, 1
1, 2, 1
-2, -4, 2, 1
8, 16, -4, 2, 1
-40, -80, 16, -4, 2, 1
224, 448, -80, 16, -4, 2, 1
-1344, -2688, 448, -80, 16, -4, 2, 1
8448, 16896, -2688, 448, -80, 16, -4, 2, 1
... which is based on A052701.

Cf. A000032, A000045, A002522, A005408, A104726, A123265

T(n,0) = T(n-1,0) + T(n-2,0), T(n,k) = T(n-1,k-1) + 2*T(n-2,k-1) for k>0.
Sum_{k, 0<=k<=n} T(n,k) = A094687(n+2).
T(2n,n) = A081567(n).

A307900 Number of functions constructed from n instances of variable x using operators + (add), * (multiply), and parentheses.

Original entry on oeis.org

1, 2, 4, 10, 24, 61, 150, 382, 964, 2452, 6307, 16379, 42989, 113965, 305035, 823632, 2241814, 6145670, 16956972, 47059076, 131279567

Offset: 1

Vladimir Reshetnikov, May 04 2019

Structurally different expressions that represent the same function of x are only counted once. So, a(n) <= A052701(n).

			For n = 1, we have only one function {x}, so a(1) = 1.
For n = 2, we have {x*x, x + x} = {x^2, 2*x}, so a(2) = 2.
For n = 3, we have {x^2*x, 2*x*x, x^2 + x, 2*x + x} = {x^3, 2*x^2, x^2 + x, 3*x}, so a(3) = 4.
For n = 4, we have {x^4, 2*x^3, x^3 + x^2, x^3 + x, 4*x^2, 3*x^2, 2*x^2 + x, 2*x^2, x^2 + 2*x, 4*x}, so a(4) = 10.

Cf. A048249, A052701.

b:= proc(n) option remember; `if`(n=1, {x}, {seq(seq(seq([f+g,
        expand(f*g)][], g=b(n-i)), f=b(i)), i=1..iquo(n, 2))})
    end:
a:= n-> nops(b(n)):
seq(a(n), n=1..12);  # Alois P. Heinz, May 04 2019

ClearAll[a, f, x, n, k]; f[1] = {x}; f[n_Integer] := f[n] = DeleteDuplicates[Expand[Flatten[Table[Outer[#1[#2, #3] &, {Times, Plus}, f[k], f[n - k]], {k, n/2}]]]]; a[n_Integer] := Length[f[n]]; Table[a[n], {n, 15}]

a(19)-a(20) from Alois P. Heinz, May 04 2019

a(21) from Vladimir Reshetnikov, May 05 2019

cp's OEIS Frontend

A192404 G.f. satisfies: A(x,y) = 1 + Sum_{n>=1} x^n*y*A(x,y)^n/(1 - y*A(x,y)^(2*n)), where A(x,y) = 1 + Sum_{n>=1,k>=1} T(n,k)*x^n*y^k; here the coefficients T(n,k) form a square array and are read by antidiagonals.

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PARI

PARI

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A234939 Coefficients of Hilbert series for suboperad of bicolored noncrossing configurations generated by a triangle with colored base and at least one more colored edge and a triangle with one colored non-base edge.

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A090375 Number of unrooted Eulerian maps with bicolored faces which are self-isomorphic under reversing the colors.

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Mathematica

PARI

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A101477 Square array T(n,k), read by antidiagonals: number of labeled trees, with increments of labels along edges constrained to +-1, with n nodes that have no label greater than k.

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A101596 G.f.: c(2*x)^4, where c(x) is the g.f. of A000108.

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PARI

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A114608 Triangle read by rows: T(n,k) is the number of bicolored Dyck paths of semilength n and having k peaks of the form ud (0 <= k <= n). A bicolored Dyck path is a Dyck path in which each up-step is of two kinds: u and U.

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Maple

Mathematica

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A117505 Triangle of coefficients for polynomials used for the column g.f.s of triangle A116880, called CM(1,2).

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A234938 Coefficients of Hilbert series for the suboperad of bicolored noncrossing configurations generated by a fully colored triangle and a fully uncolored triangle.

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Mathematica

A192404 G.f. satisfies: A(x,y) = 1 + Sum_{n>=1} x^nyA(x,y)^n/(1 - yA(x,y)^(2n)), where A(x,y) = 1 + Sum_{n>=1,k>=1} T(n,k)x^ny^k; here the coefficients T(n,k) form a square array and are read by antidiagonals.

A214582 Riordan array (1/(1-x-x^2), x(1+2x)).