A053764 -id:A053764 - 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

A060722 a(n) = 3^(n^2).

Original entry on oeis.org

1, 3, 81, 19683, 43046721, 847288609443, 150094635296999121, 239299329230617529590083, 3433683820292512484657849089281, 443426488243037769948249630619149892803

Offset: 0

Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001

The number of n X n {-1,0,1}-matrices (the same as the number of n X n matrices over GF(3) ).

Harry J. Smith, Table of n, a(n) for n = 0..45
Robert Corless and Steven Thornton, The Bohemian Eigenvalue Project, 2017 poster.
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.

Cf. A053290, A053764, A378666.

for n from 1 to 15 do printf(`%d,`,3^(n^2)) od:

Array[3^(#^2) &, 9] (* Michael De Vlieger, Jul 12 2018 *)

{ for (n=0, 45, write("b060722.txt", n, " ", 3^(n^2)); ) } \\ Harry J. Smith, Jul 10 2009

a(n) = 3^(n^2); \\ Joerg Arndt, Feb 23 2014

a(n) = [x^n] 1/(1 - 3^n*x). - Ilya Gutkovskiy, Oct 10 2017
From Geoffrey Critzer, Dec 02 2024: (Start)
a(n) = Sum_{k=0..n} A378666(n,k)*A053764(k)*A053290(n-k).
Sum_{n>=0} a(n)x^n/B(n) = f(x)*g(x) where f(x) = Sum_{n>=0} A053290(n)x^n/B(n) and g(x) = Sum_{n>=0} A053764(n)x^n/B(n) and B(n) = A053290(n)/2^n. (End)

More terms from James Sellers, Apr 24 2001

a(0) = 1 added by N. J. A. Sloane, Nov 23 2007

A109345 a(n) = 5^((n^2 - n)/2).

Original entry on oeis.org

1, 1, 5, 125, 15625, 9765625, 30517578125, 476837158203125, 37252902984619140625, 14551915228366851806640625, 28421709430404007434844970703125

Offset: 0

Philippe Deléham, Aug 21 2005

Sequence given by the Hankel transform (see A001906 for definition) of A078009 = {1, 1, 6, 41, 306, 2426, 20076, 171481, ...}; example: det([1, 1, 6, 41; 1, 6, 41, 306; 6, 41, 306, 2426; 41, 306, 2426, 20076]) = 5^6 = 15625.

a(n) is the number of simple labeled graphs, with bi-directional and non-directed edges allowed and not regarded as equivalent, on n labeled nodes. - Mark Stander, Feb 07 2019

G. C. Greubel, Table of n, a(n) for n = 0..50

Cf. A006125 (number of graphs on n labeled nodes), A047656 (number of semi-complete digraphs on n labeled nodes), A053763 (number of simple digraphs on n labeled nodes), A053764.

List([0..12], n -> 5^Binomial(n,2)); # G. C. Greubel, Feb 09 2019

[5^Binomial(n,2): n in [0..12]]; // G. C. Greubel, Feb 09 2019

seq(5^(binomial(2+n,n)), n=-2..8); # Zerinvary Lajos, Jun 12 2007

5^Binomial[Range[0, 12], 2] (* G. C. Greubel, Feb 09 2019 *)

a(n)=5^binomial(n,2) \\ Charles R Greathouse IV, Jan 11 2012

[5^binomial(n,2) for n in (0..12)] # G. C. Greubel, Feb 09 2019

a(n+1) is the determinant of n X n matrix M_(i, j) = binomial(5i, j).

G.f. A(x) satisfies: A(x) = 1 + x * A(5*x). - Ilya Gutkovskiy, Jun 04 2020

A053765 a(n) = 4^(n^2 - n).

Original entry on oeis.org

1, 1, 16, 4096, 16777216, 1099511627776, 1152921504606846976, 19342813113834066795298816, 5192296858534827628530496329220096, 22300745198530623141535718272648361505980416

Offset: 0

Stephen G Penrice, Mar 29 2000

Number of nilpotent n X n matrices over GF(4).
(-1)^n * resultant of the Chebyshev polynomial of first kind of degree n and Chebyshev polynomial of first kind of degree 2n (cf. A039991). - Benoit Cloitre, Jan 26 2003
a(n) is the number of spanning subgraphs (or equivalently sets of edges) in the n X n grid graph. - Andrew Howroyd, Jan 29 2023

N. J. Fine and I. N. Herstein, The probability that a matrix be nilpotent, Illinois J. Math., 2 (1958), 499-504.
M. Gerstenhaber, On the number of nilpotent matrices with coefficients in a finite field. Illinois J. Math., Vol. 5 (1961), 330-333.

Andrew Howroyd, Table of n, a(n) for n = 0..40

Cf. A053763, A053764.

a(n) = 4^(n^2 - n) \\ Andrew Howroyd, Jan 29 2023

More terms from James Sellers, Apr 08 2000

A052497 Number of nonsingular n X n matrices over GF(9).

Original entry on oeis.org

1, 8, 5760, 339655680, 1624314979123200, 629282246371356907929600, 19747506525777609095698646040576000, 50195501537943419769100848121708339934527488000

Offset: 0

Vladeta Jovovic, Mar 16 2000

G. C. Greubel, Table of n, a(n) for n = 0..30
Jeffrey Overbey, William Traves, and Jerzy Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29, Iss. 1 (2005), pp. 59-72; author's copy.

Cf. A027877, A053764.

Cf. A002884, A003792, A053290, A053291, A053292, A053293, A052496, A052498, A132037.

[1] cat [&*[(9^n - 9^k): k in [0..n-1]]: n in [1..10]]; // Bruno Berselli, Jan 28 2013

Table[Product[(9^n - 9^j), {j, 0, n-1}], {n, 0, 10}] (* G. C. Greubel, May 14 2019 *)

{a(n) = prod(j=0,n-1, 9^n - 9^j)}; \\ G. C. Greubel, May 14 2019

[product(9^n - 9^j for j in (0..n-1)) for n in (0..10)] # G. C. Greubel, May 14 2019

a(n) = (9^n - 1)*(9^n - 9)*...*(9^n - 9^(n-1)).
a(n) = A053764(n)*A027877(n). - Bruno Berselli, Jan 30 2013
a(n) ~ c * 9^(n^2), where c = A132037. - Amiram Eldar, Jul 06 2025

A109354 a(n) = 6^((n^2 - n)/2).

Original entry on oeis.org

1, 1, 6, 216, 46656, 60466176, 470184984576, 21936950640377856, 6140942214464815497216, 10314424798490535546171949056, 103945637534048876111514866313854976, 6285195213566005335561053533150026217291776, 2280250319867037997421842330085227917956272625811456

Offset: 0

Philippe Deléham, Aug 25 2005

nonn

Sequence given by the Hankel transform (see A001906 for definition) of A078018 = {1, 1, 7, 55, 469, 4237, 39907, 387739, ...}; example: det([1, 1, 7, 55; 1, 7, 55, 469; 7, 55, 469, 4237; 55, 469, 4237, 39907]) = 6^6 = 46656.

In general, sequences of the form m^((n^2 - n)/2) enumerate the graphs with n labeled nodes with m types of edge. a(n) therefore is the number of labeled graphs with n nodes with 6 types of edge. - Mark Stander, Apr 11 2019

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A000400, A006125, A047656, A053763, A053764, A109345.

Table[6^((n^2-n)/2),{n,0,10}] (* Harvey P. Dale, May 28 2013 *)

a(n) = 6^((n^2 - n)/2); \\ Michel Marcus, Apr 12 2019

a(n+1) is the determinant of n X n matrix M_(i, j) = binomial(6i, j).

G.f. A(x) satisfies: A(x) = 1 + x * A(6*x). - Ilya Gutkovskiy, Jun 04 2020

Terms a(11) and beyond from Andrew Howroyd, Jan 02 2020

A109493 a(n) = 7^((n^2 - n)/2).

Original entry on oeis.org

1, 1, 7, 343, 117649, 282475249, 4747561509943, 558545864083284007, 459986536544739960976801, 2651730845859653471779023381601, 107006904423598033356356300384937784807

Offset: 0

Philippe Deléham, Aug 29 2005

Sequence given by the Hankel transform (see A001906 for definition) of A081178 = {1, 1, 8, 71, 680, 6882, 72528, 788019, ...}; example: det([1, 1, 8, 71; 1, 8, 71, 680; 8, 71, 680, 6882; 71, 680, 6882, 72528]) = 7^6 = 117649.

In general, sequences of the form m^((n^2 - n)/2) enumerate the graphs with n labeled nodes with m types of edge. a(n) therefore is the number of labeled graphs with n nodes with 7 types of edge. - Mark Stander, Apr 11 2019

Cf. A006125, A047656, A053763, A053764, A109345, A109354.

a(n) = 7^binomial(n,2) \\ Charles R Greathouse IV, Jan 17 2012

a(n+1) is the determinant of n X n matrix M_(i, j) = binomial(7i, j).

G.f. A(x) satisfies: A(x) = 1 + x * A(7*x). - Ilya Gutkovskiy, Jun 04 2020

A109966 a(n) = 8^((n^2-n)/2).

Original entry on oeis.org

1, 1, 8, 512, 262144, 1073741824, 35184372088832, 9223372036854775808, 19342813113834066795298816, 324518553658426726783156020576256, 43556142965880123323311949751266331066368, 46768052394588893382517914646921056628989841375232, 401734511064747568885490523085290650630550748445698208825344

Offset: 0

Philippe Deléham, Sep 01 2005

Sequence given by the Hankel transform (see A001906 for definition) of A082147 = {1, 1, 9, 89, 945, 10577, 123129, 1476841, ...}; example: det([1, 1, 9, 89; 1, 9, 89, 945; 9, 89, 945, 10577; 89, 945, 10577, 123129]) = 8^6 = 262144.

The number of labeled multigraphs on n vertices such that (i) no self loops are allowed; (ii) all edges are painted in one of 3 colors; (iii) edges between any pair of vertices are painted in distinct colors. Note, this implies that there are at most 3 edges between any vertex pair. Also note there is no restriction on the color of edges incident to a common vertex. - Geoffrey Critzer, Jan 14 2020

Cf. A006125, A047656, A053763, A053764, A059435, A082147, A109345, A109354, A109493.

[2^(3*Binomial(n,2)): n in [0..10]]; // G. C. Greubel, Feb 05 2018

Table[2^(3*Binomial[n,2]),{n,0,10}] (* Geoffrey Critzer, Nov 10 2011 *)

a(n)=8^binomial(n,2) \\ Charles R Greathouse IV, Jan 17 2012

a(n+1) is the determinant of n X n matrix M_(i, j) = binomial(8i, j).

Hankel transform of A059435. - Philippe Deléham, Sep 03 2006

a(10) corrected and a(11), a(12) from Georg Fischer, Apr 01 2022

A053854 Number of n X n matrices over GF(3) of order dividing 9, i.e., the number of solutions to X^9=I in GL(n,3).

Original entry on oeis.org

1, 9, 729, 531441, 3486784401, 205891132094649, 109418989131512359209, 523347633027360537213511521, 22528399544939174411840147874772641, 1394761471471951120984262893478242219427049, 601851824520496078935516587103606691779438596774649

Offset: 1

Vladeta Jovovic, Mar 28 2000

nonn

Is this the same sequence (apart from the initial term) as A053764? - Philippe Deléham, Dec 09 2007
From M. F. Hasler, Oct 14 2008: (Start)
X^9 = I <=> I - X^9 = 0 <=> (I - X)^9 = 0 in GF(3). So to any solution of the first equation corresponds a solution X' = I-X of the other equation and vice versa. On the other hand, from considerations about the matrix rank (e.g., reasoning in Jordan basis) it is known that to check for nilpotency it is sufficient to go up to an exponent equal to the size of the matrix.
Thus by going out to the 9th power one finds all nilpotent matrices for sizes <= 9 X 9. Since A053854 is only given up to n=9, we can't see if A053764(10) is strictly bigger than A053854(10), which seems very likely since from then on there should be more matrices that satisfy A^10=0 than there are matrices satisfying A^9=0. (End)

V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Cf. A053774.

More terms from Sean A. Irvine, Jan 16 2022

A110147 10^((n^2-n)/2).

Original entry on oeis.org

1, 1, 10, 1000, 1000000, 10000000000, 1000000000000000, 1000000000000000000000, 10000000000000000000000000000, 1000000000000000000000000000000000000

Offset: 0

Philippe Deléham, Sep 04 2005

Sequence given by the Hankel transform (see A001906 for definition) of A082148 = {1, 1, 11, 131, 1661, 22101, 305151, 4335711, ...}; example: det([1, 1, 11, 131; 1, 11, 131, 1661; 11, 131, 1661, 22101; 131, 1661, 22101, 305151]) = 10^6 = 1000000.

Also the Hankel transform of A379103. - Nathaniel Johnston, Dec 16 2024

Cf. A006125, A047656, A053763, A053764, A109345, A109354, A109493, A109966.

a(n)=10^binomial(n,2) \\ Charles R Greathouse IV, Jan 17 2012

a(n+1) is the determinant of n X n matrix M_(i, j) = binomial(10i, j).

a(n)=10a(n-1)^2/a(n-2), a(0)=a(1)=1. - Michael Somos, Sep 12 2005

cp's OEIS Frontend

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

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A060722 a(n) = 3^(n^2).

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A109345 a(n) = 5^((n^2 - n)/2).

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A053765 a(n) = 4^(n^2 - n).

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A052497 Number of nonsingular n X n matrices over GF(9).

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A109354 a(n) = 6^((n^2 - n)/2).

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