A003645 -id:A003645 - cp's OEIS frontend

A369502 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2+x)^2 ).

1, 6, 47, 420, 4059, 41316, 436345, 4737018, 52535950, 592667532, 6779699073, 78458218746, 916886214115, 10805128064100, 128260666769895, 1532180536574580, 18405744106135914, 222204347510440092, 2694506677864591810, 32804976554127379680, 400837173223351237295

Offset: 0

Seiichi Manyama, Jan 25 2024

nonn

Index entries for reversions of series

Cf. A369503, A369504.

Cf. A003645, A369505.

my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^2+x)^2)/x)

a(n) = sum(k=0, n, binomial(2*n+2, k)*binomial(4*n-2*k+4, n-k))/(n+1);

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

A069725 Number of nonisomorphic unrooted unicursal planar maps with n edges and two vertices of valency 1 (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

Original entry on oeis.org

1, 1, 3, 11, 62, 342, 2152, 13768, 91800, 622616, 4301792, 30100448, 213019072, 1521473984, 10954616064, 79420280064, 579300888960, 4248201302400, 31302536066560, 231638727063040

Offset: 1

Valery A. Liskovets, Apr 07 2002

There is an easy formula.

V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

Cf. A069727, A003645, A069732.

a[1] = a[2] = 1;
a[n_] := With[{m = Floor[(n+1)/2]}, 1/n 2^(n-3) Binomial[2n-2, n-1] + 2^(m-3) Binomial[2m-2, m-1]];
Array[a, 20] (* Jean-François Alcover, Aug 28 2019 *)

A069731 Number of unicursal planar maps with n edges rooted at a vertex of odd valency (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

Original entry on oeis.org

1, 5, 28, 168, 1056, 6864, 45760, 311168, 2149888, 15049216, 106502144, 760729600, 5477253120, 39710085120, 289650032640, 2124100239360, 15651264921600, 115819360419840, 860372391690240

Offset: 1

Valery A. Liskovets, Apr 07 2002

V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

Cf. A069724, A069720, A000108, A003645.

Cf. A000346, A001006, A136576.

Z:=-(1-4*z-sqrt(1-4*z))/sqrt(1-4*z)/64: Zser:=series(Z, z=0, 32): seq(coeff(Zser*2^(n+1), z, n), n=3..24); # Zerinvary Lajos, Jan 01 2007

Table[2^(n-2) CatalanNumber[n+1], {n, 1, 19}] (* Jean-François Alcover, Aug 28 2019 *)

a(n) = 2^(n-2)*C_(n+1), where C_n stands for the Catalan numbers (A000108).
a(n) = A003645(n+2)/4.
D-finite with recurrence: 4*(2*n+1)*a(n-1) - (n+2)*a(n) = 0, a(1) = 1. - Georg Fischer, May 23 2021
From Peter Bala, Apr 29 2024: (Start)
a(n) = Sum_{k = 0..n} binomial(n, 2*k)*Catalan(k)*4^(n-k-1).
O.g.f.: A(x) = (1 - 4*x - 8*x^2 - sqrt(1 - 8*x))/(32*x^2).
A(x) = series reversion of x*c(-x)/(1 + 4*x), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108 and c(-x)/(1 + 4*x) is the g.f. of (-1)^n*A000346(n). (End)

A167432 Riordan array (c(2x)^2,xc(2x)), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 4, 1, 20, 6, 1, 112, 36, 8, 1, 672, 224, 56, 10, 1, 4224, 1440, 384, 80, 12, 1, 27456, 9504, 2640, 600, 108, 14, 1, 183040, 64064, 18304, 4400, 880, 140, 16, 1, 1244672, 439296, 128128, 32032, 6864, 1232, 176, 18, 1, 8599552, 3055104, 905216, 232960

Offset: 0

Paul Barry, Nov 03 2009

Inverse of (1-4x+4x^2,x(1-2x)) (A167431). Row sums are A084076. First column is A003645.

			Triangle begins
1,
4, 1,
20, 6, 1,
112, 36, 8, 1,
672, 224, 56, 10, 1,
4224, 1440, 384, 80, 12, 1,
27456, 9504, 2640, 600, 108, 14, 1,
183040, 64064, 18304, 4400, 880, 140, 16, 1,
1244672, 439296, 128128, 32032, 6864, 1232, 176, 18, 1,
8599552, 3055104, 905216, 232960, 52416, 10192, 1664, 216, 20, 1,
60196864, 21498880, 6449664, 1697280, 396032, 81536, 14560, 2184, 260, 22, 1
The production matrix is
4, 1,
4, 2, 1,
8, 4, 2, 1,
16, 8, 4, 2, 1,
32, 16, 8, 4, 2, 1,
64, 32, 16, 8, 4, 2, 1,
128, 64, 32, 16, 8, 4, 2, 1,
256, 128, 64, 32, 16, 8, 4, 2, 1,
512, 256, 128, 64, 32, 16, 8, 4, 2, 1
When topped with the row (1,0,0,0...), this has inverse
1,
-4, 1,
4, -2, 1,
0, 0, -2, 1,
0, 0, 0, -2, 1,
0, 0, 0, 0, -2, 1,
0, 0, 0, 0, 0, -2, 1,
0, 0, 0, 0, 0, 0, -2, 1,
0, 0, 0, 0, 0, 0, 0, -2, 1,
0, 0, 0, 0, 0, 0, 0, 0, -2, 1

Number triangle T(n,k)=A054445(n,k)*2^(n-k).

A181282 a(n) is the number of associate Rota-Baxter words in one idempotent generator x and one idempotent operator P of degree n. Such words are Rota-Baxter words that begin and/or end with x, and P is applied n times in the word.

Original entry on oeis.org

1, 3, 12, 60, 336, 2016, 12672, 82368, 549120, 3734016, 25798656, 180590592, 1278025728, 9128755200, 65727037440, 476521021440, 3475800391680, 25489202872320, 187815179059200, 1389832325038080, 10324468700282880

Offset: 0

William Sit (wyscc(AT)sci.ccny.cuny.edu), Oct 11 2010

nonn

			For n = 2, the a(2) = 12 associate Rota-Baxter words are: xP(xP(x)), xP(xP(x))x, P(xP(x))x, xP(P(x)x), xP(P(x)x)x, P(P(x)x)x, xP(xP(x)x), xP(xP(x)x)x, P(xP(x)x)x, xP(x)xP(x), xP(x)xP(x)x, P(x)xP(x)x.

Indranil Ghosh, Table of n, a(n) for n = 0..1000
L. Guo and W. Sit, Enumeration and generating functions of Rota-Baxter words, Math. Comp. Sci. (2010). In G. Regensburger, M. Rosenkranz, and W. Sit, eds., Algebraic and Algorithmic Aspects of Differential and Integral Operators (AADIOS), Sp. Issue, Math. C. Sc., 4 (2,3) (2010).

Cf. A000108, A003645, A025225, A000257.

[1] cat [3*2^(n-1)*Catalan(n): n in [1..40]]; // G. C. Greubel, Jan 04 2023

CoefficientList[Series[(3-4x-3Sqrt[1-8x])/(8x), {x,0,40}], x]
a[0] = 1; a[n_]:= 3*2^(n-1) CatalanNumber[n]; Table[a[n], {n,0,20}] (* Indranil Ghosh, Mar 05 2017 *)

a(n) = if(n==0, 1, 3*2^(n-1)*(binomial(2*n,n)/(n+1))); \\ Indranil Ghosh, Mar 05 2017

import math
f = math.factorial
def C(n,r): return f(n)/f(r)/f(n-r)
def A181282(n): return 1 if n==0 else 3*2**(n-1)*(C(2*n,n)/(n+1)) # Indranil Ghosh, Mar 05 2017

[3*2^(n-1)*catalan_number(n) -int(n==0)/2 for n in range(41)] # G. C. Greubel, Jan 04 2023

a(n) = 3*2^(n-1)*A000108(n).
G.f.: (3 - 4*t - 3*sqrt(1-8*t))/(8*t).
(n+1)*a(n) = 4*(2*n-1)*a(n-1). - R. J. Mathar, Jul 24 2012
a(n) = (n+2) * A000257(n). - F. Chapoton, Feb 26 2024

A069732 Number of nonisomorphic unrooted unicursal planar maps with n edges and no vertices of valency 1 (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

Original entry on oeis.org

0, 0, 2, 7, 40, 239, 1549, 10396, 71467, 498598, 3520015, 25087426, 180249182, 1304148015, 9494015372, 69492950976, 511147940104, 3776180492129, 28007532925171, 208474866181148

Offset: 1

Valery A. Liskovets, Apr 07 2002

V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

Cf. A069724, A003645, A069725, A005470.

A003645[n_] := 2^n CatalanNumber[n + 1];
A069724[n_] := 1/(2 n) DivisorSum[n, If[OddQ[n/#], EulerPhi[n/#] 2^(# - 2) Binomial[2 #, #], 0] &] + If[OddQ[n], 2^((n - 3)/2) Binomial[n - 1, (n - 1)/2], 2^((n - 6)/2) Binomial[n, n/2]];
A069725[n_] := If[n <= 2, 1, With[{m = Floor[(n + 1)/2]}, 1/n 2^(n - 3) Binomial[2 n - 2, n - 1] + 2^(m - 3) Binomial[2 m - 2, m - 1]]];
a[n_] := If[n == 1, 0, A069724[n] - A003645[n - 2] - A069725[n]];
Array[a, 20] (* Jean-François Alcover, Aug 28 2019 *)

a(n) = A069724(n) - A003645(n) - A069725(n).

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

A123382 Triangle T(n,k), 0 <= k <= n, defined by : T(n,k) = 0 if k < 0, T(0,k) = 0^k, (n+2)(2n-2k+1)T(n,k) = (2n+1)( 4(2n-2k+1)T(n-1,k-1) + (n+2k+2)T(n-1,k) ).

Original entry on oeis.org

1, 1, 4, 1, 15, 20, 1, 35, 168, 112, 1, 66, 714, 1680, 672, 1, 110, 2178, 11352, 15840, 4224, 1, 169, 5434, 51051, 156156, 144144, 27456, 1, 245, 11830, 178035, 972400, 1953952, 1281280, 183040, 1, 340, 23324, 520676, 4516798, 16102944, 22870848, 11202048, 1244672, 1, 456, 42636, 1337220, 17073134

Offset: 0

Philippe Deléham, Oct 13 2006

G. Kreweras explains that since the rows of A140136 are symmetric, they can be considered as linear combinations of the odd-indexed rows of the Pascal triangle. For instance, (1,1) = 1*(1,1) and (1,7,7,1) = 1*(1,3,3,1) + 4*(0,1,1,0) and (1,20,75,75,10,1) = 1*(1,5,10,10,5,1) + 15*(0,1,3,3,1) + 20*(0,0,1,1,0,0). These coefficients (1; 1, 4; 1, 15, 20;) are the rows of this triangle. - Michel Marcus, Nov 17 2014

			Triangle begins:
0: 1;
1: 1, 4;
2: 1, 15, 20;
3: 1, 35, 168, 112;
4: 1, 66, 714, 1680, 672;
5: 1, 110, 2178, 11352, 15840, 4224;
6: 1, 169, 5434, 51051, 156156, 144144, 27456;
7: 1, 245, 11830, 178035, 972400, 1953952, 1281280, 183040;
8: 1, 340, 23324, 520676, 4516798, 16102944, 22870848, 11202048, 1244672;
.....

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Germain Kreweras, Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, 6 (1965).
Germain Kreweras, page 93 of "Sur une classe de problèmes de dénombrement...", containing the defining formula for this sequence.

T[0, 0] := 1; T[0, k_] := 0; T[n_, k_] := T[n, k] = (2*n + 1)*(4*(2*n - 2*k + 1)*T[n - 1, k - 1] + (n + 2*k + 2)*T[n - 1, k])/((n + 2)*(2*n - 2*k + 1)); Table[If[k < 0, 0, T[n, k]], {n, 0, 5}, {k, 0, n}]//Flatten (* G. C. Greubel, Oct 13 2017 *)

@CachedFunction
def T(n,k):
    if k < 0: return 0
    if n < 0: return 0
    if n == 0: return int( k==0 )
    if k == 0: return 1
    return ( (2*n+1)*( 4*(2*n-2*k+1)*T(n-1,k-1) + (n+2*k+2)*T(n-1,k) ) ) / ((n+2)*(2*n-2*k+1))
for n in [0..16]:
    print([T(n,k) for k in range(0,n+1)])
# Joerg Arndt, Nov 21 2014

T(n,n) = A003645(n).

Corrected name, added more terms, Joerg Arndt, Nov 21 2014

A369505 Expansion of (1/x) * Series_Reversion( x / ((1+x)^3+x)^2 ).

Original entry on oeis.org

1, 8, 86, 1066, 14361, 204314, 3020745, 45955442, 714723588, 11312450432, 181625888244, 2950848879096, 48423670556100, 801454908292020, 13363137183238881, 224253208102065664, 3784736105491395780, 64197997357038408976, 1093863031541592651003

Offset: 0

Seiichi Manyama, Jan 25 2024

nonn

Index entries for reversions of series

Cf. A003645, A369502.

my(N=20, x='x+O('x^N)); Vec(serreverse(x/((1+x)^3+x)^2)/x)

a(n) = sum(k=0, n, binomial(2*n+2, k)*binomial(6*n-3*k+6, n-k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(2*n+2,k) * binomial(6*n-3*k+6,n-k).

A134183 A Hankel transform of a Catalan product.

Original entry on oeis.org

0, 1, 16, 768, 131072, 83886080, 206158430208, 1970324836974592, 73786976294838206464, 10880332376531662572355584, 6338253001141147007483516026880

Offset: 0

Paul Barry, Oct 11 2007

-a(n) is the Hankel transform of A000108(n)*(2^n + 0^n)/2 = A003645(n-1). The Hankel transform of A003645(n) is 4^binomial(n+1, 2).

a(n) = 4^binomial(n,2)*n*2^(n-1);

a(n) = A053763(n)*A001787(n).

cp's OEIS Frontend

A369502 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2+x)^2 ).

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A069725 Number of nonisomorphic unrooted unicursal planar maps with n edges and two vertices of valency 1 (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

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A069731 Number of unicursal planar maps with n edges rooted at a vertex of odd valency (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

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A167432 Riordan array (c(2x)^2,xc(2x)), c(x) the g.f. of A000108.

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A181282 a(n) is the number of associate Rota-Baxter words in one idempotent generator x and one idempotent operator P of degree n. Such words are Rota-Baxter words that begin and/or end with x, and P is applied n times in the word.

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A069732 Number of nonisomorphic unrooted unicursal planar maps with n edges and no vertices of valency 1 (unicursal means that exactly two vertices are of odd valency; there is an Eulerian path).

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

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A123382 Triangle T(n,k), 0 <= k <= n, defined by : T(n,k) = 0 if k < 0, T(0,k) = 0^k, (n+2)*(2*n-2*k+1)*T(n,k) = (2*n+1)*( 4*(2*n-2*k+1)*T(n-1,k-1) + (n+2*k+2)*T(n-1,k) ).

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A123382 Triangle T(n,k), 0 <= k <= n, defined by : T(n,k) = 0 if k < 0, T(0,k) = 0^k, (n+2)(2n-2k+1)T(n,k) = (2n+1)( 4(2n-2k+1)T(n-1,k-1) + (n+2k+2)T(n-1,k) ).