A025226 -id:A025226 - cp's OEIS frontend

A005159 a(n) = 3^n*Catalan(n).

1, 3, 18, 135, 1134, 10206, 96228, 938223, 9382230, 95698746, 991787004, 10413763542, 110546105292, 1184422556700, 12791763612360, 139110429284415, 1522031755700070, 16742349312700770, 185047018719324300, 2054021907784499730

Offset: 0

N. J. A. Sloane, Valery A. Liskovets

Total number of vertices in rooted planar maps with n edges.
Number of blossom trees with n inner vertices.
The number of rooted n-edge maps in the plane (planar with a distinguished outside face). - Valery A. Liskovets, Mar 17 2005
Hankel transform is 3^(n+n^2) = A053764(n+1). - Philippe Deléham, Dec 10 2007
From Joerg Arndt, Oct 22 2012: (Start)
Also the number of strings of length 2*n of three different types of balanced parentheses.
The number of strings of length 2*n of t different types of balanced parentheses is given by t^n * A000108(n): there are n opening parentheses in the strings, giving t^n choices for the type (the closing parentheses are chosen to match). (End)
Number of Dyck paths of length 2n in which the step U=(1,1) come in 3 colors. - José Luis Ramírez Ramírez, Jan 31 2013
Number of unknown entries in bracketed Kleene's truth table connected by the implication with n distinct variables. See Yildiz link. - Michel Marcus, Oct 21 2020

Leonid M. Koganov, Valery A. Liskovets and Timothy R. S. Walsh, Total vertex enumeration in rooted planar maps, Ars Combin., Vol. 54 (2000), pp. 149-160.
Valery A. Liskovets and Timothy R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.
Richard P. Stanley, Catalan Numbers, Cambridge, 2015, p. 107.

T. D. Noe, Table of n, a(n) for n = 0..100
Mireille Bousquet-Mélou, Limit laws for embedded trees, arXiv:math/0501266 [math.CO], 2005.
J. Bouttier, P. Di Francesco and E. Guitter, Statistics of planar graphs viewed from a vertex: a study via labeled trees, Nucl. Phys. B, Vol. 675, No. 3 (2003), pp. 631-660. See p. 631, eq. (3.3).
Zhi Chen and Hao Pan, Identities involving weighted Catalan-Schroder and Motzkin Paths, arXiv:1608.02448 [math.CO], 2016. See eq. (1.13), a=b=3.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Gerard 't Hooft, Counting planar diagrams with various restrictions, Nucl. Phys. B, Vol. 538, No. 1-2 (1999), pp. 389-410; arXiv:hep preprint arXiv:hep-th/9808113, 1998.
Hsien-Kuei Hwang, Mihyun Kang and Guan-Huei Duh, Asymptotic Expansions for Sub-Critical Lagrangean Forms, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.
Sergey Kitaev, Anna de Mier and Marc Noy, On the number of self-dual rooted maps, European J. Combin., Vol. 35 (2014), pp. 377-387. MR3090510.
Valery A. Liskovets, A pattern of asymptotic vertex valency distributions in planar maps, J. Combin. Th. B, Vol. 75, No. 1 (1999), pp. 116-133.
Valery A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., Vol. 36, No. 4 (2006), pp. 364-387.
Thomas M. Richardson, The three 'R's and Dual Riordan Arrays, arXiv:1609.01193 [math.CO], 2016.
Gilles Schaeffer and Paul Zinn-Justin, On the asymptotic number of plane curves and alternating knots, arXiv:math-ph/0304034, 2003-2004.
Simeon T. Stefanov, Counting fixed points free vector fields on B^2, arXiv:1807.03714 [math.GT], 2018.
Volkan Yildiz, Counting with 3-valued truth tables of bracketed formulae connected by implication, arXiv:2010.10303 [math.GM], 2020.

Cf. A000108, A000244, A025226, A053764, A085880.

Limit of array A102994.

List([0..20],n->3^n*Binomial(2*n,n)/(n+1)); # Muniru A Asiru, Mar 30 2018

[3^n*Catalan(n): n in [0..20]]; // Vincenzo Librandi, Oct 16 2018

A005159_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := 3*(a[w-1]+add(a[j]*a[w-j-1],j=1..w-1)) od;convert(a,list)end: A005159_list(19); # Peter Luschny, May 19 2011

InverseSeries[Series[y-3*y^2, {y, 0, 24}], x] (* then A(x)=y(x)/x *) (* Len Smiley, Apr 07 2000 *)
Table[3^n CatalanNumber[n],{n,0,30}] (* Harvey P. Dale, May 18 2011 *)
CoefficientList[Series[(1 - Sqrt[1-4*(3*x)])/(6*x), {x, 0, 50}], x] (* Stefano Spezia, Oct 16 2018 *)

a(n) = 3^n*binomial(2*n,n)/(n+1) \\ Charles R Greathouse IV, Feb 06 2017

G.f.: 2/(1+sqrt(1-12x)) = (1 - sqrt(1-4*(3*x))) / (6*x).
With offset 1 : a(1)=1, a(n) = 3*Sum_{i=1..n-1} a(i)*a(n-i). - Benoit Cloitre, Mar 16 2004
G.f.: c(3*x) with c(x) the o.g.f. of A000108 (Catalan).
From Gary W. Adamson, Jul 12 2011: (Start)
a(n) is the upper left term in M^n, M = the infinite square production matrix:
  3, 3, 0, 0, 0, 0, ...
  3, 3, 3, 0, 0, 0, ...
  3, 3, 3, 3, 0, 0, ...
  3, 3, 3, 3, 3, 0, ...
  3, 3, 3, 3, 3, 3, ...
  ... (End)
D-finite with recurrence (n+1)*a(n)+6*(1-2n)*a(n-1)=0. - R. J. Mathar, Apr 01 2012
E.g.f.: a(n) = n!* [x^n] KummerM(1/2, 2, 12*x). - Peter Luschny, Aug 25 2012
a(n) = sum_{k=0..n} A085880(n,k)*2^k. - Philippe Deléham, Nov 15 2013
From Ilya Gutkovskiy, Dec 04 2016: (Start)
E.g.f.: (BesselI(0,6*x) - BesselI(1,6*x))*exp(6*x).
a(n) ~ 12^n/(sqrt(Pi)*n^(3/2)). (End)
a(n) = A000244(n)*A000108(n). - Omar E. Pol, Mar 30 2018
Sum_{n>=0} 1/a(n) = 150/121 + 216*arctan(1/sqrt(11)) / (121*sqrt(11)). - Vaclav Kotesovec, Nov 23 2021
Sum_{n>=0} (-1)^n/a(n) = 138/169 - 216*arctanh(1/sqrt(13)) / (169*sqrt(13)). - Amiram Eldar, Jan 25 2022
G.f.: 1/(1-3*x/(1-3*x/(1-3*x/(1-3*x/(1-3*x/(1-3*x/(1-3*x/(1-3*x/(1-...))))))))) (continued fraction). - Nikolaos Pantelidis, Nov 20 2022

A108735 Expansion of sqrt(1 + 12*x).

Original entry on oeis.org

1, 6, -18, 108, -810, 6804, -61236, 577368, -5629338, 56293380, -574192476, 5950722024, -62482581252, 663276631752, -7106535340200, 76750581674160, -834662575706490, 9132190534200420, -100454095876204620, 1110282112315945800, -12324131446706998380

Offset: 0

N. J. A. Sloane, Jun 22 2005

This is also the expansion of sqrt(3)*(2*B2inv(x) - 1), where B2inv is the compositional inverse of the Bernoulli polynomial B(2, x) = 1/6 - x + x^2 = (x - 1/2)^2 - 1/12, for x >= 1/2. (see A196838 and A196839 for the Bernoulli polynomials). - Wolfdieter Lang, Aug 26 2015

			G.f. = 1 + 6*x - 18*x^2 + 108*x^3 - 810*x^4 + 6804*x^5 - 61236*x^6 + ...

Robert Israel, Table of n, a(n) for n = 0..838

Cf. A000108, A025226.

[1] cat [(2/3)*(-3)^(n+1)*Catalan(n-1): n in [1..30]]; // G. C. Greubel, May 21 2022

f:= proc(n) option remember; (18/n - 12)*procname(n-1) end proc: f(0):= 1:
map(f, [$0..100]); # Robert Israel, Aug 27 2015

CoefficientList[Series[(1 + 12 x)^(1/2), {x, 0, 19}], x] (* Michael De Vlieger, Aug 26 2015 *)
Join[{1}, RecurrenceTable[{a[1] == 6, a[n] == a[n-1] (18/n - 12)}, a, {n, 20}]] (* Vincenzo Librandi, Aug 27 2015 *)

my(x = xx+O(xx^30)); Vec(sqrt(1 + 12*x)) \\ Michel Marcus, Aug 26 2015

[(2/3)*(-3)^(n+1)*catalan_number(n-1) for n in (0..30)] # G. C. Greubel, May 21 2022

From Wolfdieter Lang, Aug 26 2015: (Start)
G.f.: sqrt(1 + 12*x) = 1 + 6*x*c(-3*x), with the g.f. c of the Catalan numbers A000108.
a(n) = -2*(-3)^n*C(n-1), n >= 1, and a(0) = 1, with C(n) = A000108(n). (End)
From Robert Israel, Aug 27 2015: (Start)
D-finite with recurrence: a(n) = (18/n - 12)*a(n-1).
a(n) ~ (-1)^(n+1)*12^n/(2*sqrt(Pi)*n^(3/2)). (End)
0 = a(n)*(144*a(n+1) +30*a(n+2)) +a(n+1)*(-6*a(n+1) +a(n+2)) for all n in Z. - Michael Somos, Aug 27 2015
a(n) = 2*(-1)^(n+1)*A025226(n). - R. J. Mathar, Jan 23 2020
From Amiram Eldar, May 28 2022: (Start)
Sum_{n>=0} 1/a(n) = (192 - 36*arcsinh(1/(2*sqrt(3)))/sqrt(13))/169.
Sum_{n>=0} (-1)^n/a(n) = (96 - 36*arcsin(1/(2*sqrt(3)))/sqrt(11))/121. (End)

A100239 G.f. A(x) satisfies: 3^n + 1 = Sum_{k=0..n} [x^k]A(x)^n and also satisfies: (3+z)^n + (1+z)^n - z^n = Sum_{k=0..n} [x^k](A(x) + z*x)^n for all z, where [x^k]A(x)^n denotes the coefficient of x^k in A(x)^n.

Original entry on oeis.org

1, 3, -3, 9, -36, 162, -783, 3969, -20817, 112023, -615033, 3431403, -19398690, 110880900, -639730305, 3720657807, -21790419444, 128398625658, -760668489729, 4528069760691, -27070491820644, 162464919528222, -978463778897637, 5911727071716891, -35821932198013809

Offset: 0

Paul D. Hanna, Nov 30 2004

sign

			From the table of powers of A(x), we see that
3^n+1 = Sum of coefficients [x^0] through [x^n] in A(x)^n:
A^1 = [1,  3], -3,    9,  -36,  162, -783, 3969, -20817, 112023, ...
A^2 = [1,  6,   3],   0,   -9,   54, -297, 1620,  -8910,  49572, ...
A^3 = [1,  9,  18,    0],   0,    0,  -27,  243,  -1701,  10935, ...
A^4 = [1, 12,  42,   36,   -9],   0,    0,    0,    -81,    972, ...
A^5 = [1, 15,  75,  135,   45,  -27],   0,    0,      0,      0, ...
A^6 = [1, 18, 117,  324,  324,    0,  -54],   0,      0,      0, ...
A^7 = [1, 21, 168,  630, 1071,  567, -189,  -81],     0,      0, ...
A^8 = [1, 24, 228, 1080, 2610, 2808,  540, -648,    -81],     0, ...
the main diagonal of which is:
[x^n]A(x)^(n+1) = (n+1)*A057083(n) for n>=0.

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

Cf. A057083, A100226, A100239, A107264.

a[n_]:= a[n]= 3^n*Boole[n<2] + 3*(-1)^(n+1)*Sum[Binomial[k+1, n-k-1]*Binomial[n-2,k]*3^k/(k+1), {k,0,n-2}];
Table[a[n], {n,0,40}] (* G. C. Greubel, May 21 2022 *)

a(n)=if(n==0, 1, (3^n+1-sum(k=0, n, polcoeff(sum(j=0, min(k, n-1), a(j)*x^j)^n + x*O(x^k), k)))/n)

a(n)=polcoeff((1+3*x+sqrt(1+6*x-3*x^2+x^2*O(x^n)))/2,n)

def A100239_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+3*x+sqrt(1+6*x-3*x^2))/2 ).list()
A100239_list(40) # G. C. Greubel, May 21 2022

G.f.: A(x) = (1+3*x+sqrt(1+6*x-3*x^2))/2.
Given g.f. A(x), then B(x) = A(x) - (1+2*x) series reversion is -B(-x). - Michael Somos, Sep 07 2005
Given g.f. A(x) and C(x) = g.f. of A025226, then B(x)=A(x)-1-2x satisfies B(x) = x - C(x*B(x)). - Michael Somos, Sep 07 2005
a(n) = 3^n*[n<2] + 3*(-1)^(n+1)*A107264(n-2). - G. C. Greubel, May 21 2022

A345189 Number of rows with the value "false" in the Kleene truth tables of all bracketed formulae with n distinct propositions p1, ..., pn connected by the binary connective of implication.

Original entry on oeis.org

1, 1, 6, 41, 330, 2882, 26604, 255313, 2521986, 25473638, 261898548, 2731724778, 28836047844, 307477681188, 3306988334808, 35833139582529, 390803960909106, 4286644113507902, 47258491871201508, 523372307883323566, 5819831138546794860, 64954314678710555612, 727371707764232349672

Offset: 1

Michel Marcus, Jun 10 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 1..925
Volkan Yildiz, Notes on algebraic structure of truth tables of bracketed formulae connected by implications, arXiv:2106.04728 [math.CO], 2021.

Cf. A005159 (unknown rows, shifted), A025226 (all rows), A345190 (true rows).

CoefficientList[Series[(-2 -Sqrt[1-12*x] +Sqrt[5 +24*x +4*Sqrt[1-12*x]])/6, {x, 0, 40}], x]//Rest (* G. C. Greubel, May 20 2022 *)

my(x='x+O('x^30)); Vec((-2-sqrt(1-12*x)+sqrt(5+24*x+4*sqrt(1-12*x)))/6)

def A345189_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (-2-sqrt(1-12*x)+sqrt(5+24*x+4*sqrt(1-12*x)))/6 ).list()
a=A345189_list(40); a[1:] # G. C. Greubel, May 20 2022

G.f.: (-2-sqrt(1-12*x)+sqrt(5+24*x+4*sqrt(1-12*x)))/6.

a(n) = 2*A005159(n-1) - A345190(n). - G. C. Greubel, May 20 2022

A345190 Number of rows with the value "true" in the Kleene truth tables of all bracketed formulae with n distinct propositions p1, ..., pn connected by the binary connective of implication.

Original entry on oeis.org

1, 5, 30, 229, 1938, 17530, 165852, 1621133, 16242474, 165923854, 1721675460, 18095802306, 192256162740, 2061367432212, 22276538889912, 242387718986301, 2653259550491034, 29198054511893638, 322835545567447092, 3584671507685675894, 39955514234936341980, 446897274497509974508

Offset: 1

Michel Marcus, Jun 10 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 1..925
Volkan Yildiz, Notes on algebraic structure of truth tables of bracketed formulae connected by implications, arXiv:2106.04728 [math.CO], 2021.

Cf. A005159 (unknown rows, shifted), A025226 (all rows), A345189 (false rows).

CoefficientList[Series[(4 -Sqrt[1-12*x] -Sqrt[5 +24*x +4*Sqrt[1-12*x]])/6, {x, 0, 40}], x]//Rest (* G. C. Greubel, May 20 2022 *)

my(x='x+O('x^30)); Vec((4-sqrt(1-12*x)-sqrt(5+24*x+4*sqrt(1-12*x)))/6)

def A345190_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (4-sqrt(1-12*x)-sqrt(5+24*x+4*sqrt(1-12*x)))/6 ).list()
a=A345190_list(40); a[1:] # G. C. Greubel, May 20 2022

G.f.: (4-sqrt(1-12*x)-sqrt(5+24*x+4*sqrt(1-12*x)))/6.

a(n) = 2*A005159(n-1) - A345189(n). - G. C. Greubel, May 20 2022

cp's OEIS Frontend

A005159 a(n) = 3^n*Catalan(n).

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A108735 Expansion of sqrt(1 + 12*x).

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A100239 G.f. A(x) satisfies: 3^n + 1 = Sum_{k=0..n} [x^k]A(x)^n and also satisfies: (3+z)^n + (1+z)^n - z^n = Sum_{k=0..n} [x^k](A(x) + z*x)^n for all z, where [x^k]A(x)^n denotes the coefficient of x^k in A(x)^n.

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A345189 Number of rows with the value "false" in the Kleene truth tables of all bracketed formulae with n distinct propositions p1, ..., pn connected by the binary connective of implication.

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A345190 Number of rows with the value "true" in the Kleene truth tables of all bracketed formulae with n distinct propositions p1, ..., pn connected by the binary connective of implication.

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