A060722 -id:A060722 - cp's OEIS frontend

A064063 Generalized Catalan numbers C(3; n).

1, 1, 4, 25, 190, 1606, 14506, 137089, 1338790, 13403950, 136846144, 1419257434, 14911016596, 158363649640, 1697452010230, 18338919413425, 199496184219910, 2183299541440150, 24021874198331080, 265559590979820910, 2948253066186839140, 32857382497018933060

Offset: 0

Wolfdieter Lang, Sep 13 2001

a(n+1) = Y_{n}(n+1) = Z_{n}, n >= 0, in the Derrida et al. 1992 reference (see A064094) for alpha=3, beta =1 (or alpha=1, beta=3).

Hankel transform is A060722. - Paul Barry, Jan 30 2009

S. Ramanujan, Modular Equations and Approximations to pi, pp. 23-39 of Collected Papers of Srinivasa Ramanujan, Ed. G. H. Hardy et al., AMS Chelsea 2000. See page 39, equation (50).

Fung Lam, Table of n, a(n) for n = 0..925
J. Abate and W. Whitt, Brownian Motion and the Generalized Catalan Numbers, J. Int. Seq. 14 (2011) # 11.2.6, example section 3.
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017).

Cf. A064062 (C(2; n)).

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 6/(5+Sqrt(1-12*x)) )); // G. C. Greubel, May 02 2019

CoefficientList[Series[6/(5+Sqrt[1-12 x]),{x,0,50}],x]  (* Harvey P. Dale, Mar 11 2011 *)

a(n)=if(n<0,0,polcoeff(serreverse((x-2*x^2)/(1+x)^2+O(x^(n+1))),n)) \\ Ralf Stephan, Jun 12 2004

{a(n)= if(n<1, n==0, polcoeff( serreverse( (x-2*x^2)/ (1+x)^2 +x*O(x^n)), n))} /* Michael Somos, Apr 11 2007 */

def a(n):
    if n==0: return 1
    return hypergeometric([1-n, n], [-n], 3).simplify()
[a(n) for n in range(24)] # Peter Luschny, Nov 30 2014

G.f.: (1+3*x*c(3*x)/2)/(1+x/2) = 1/(1-x*c(3*x)) with c(x) g.f. of Catalan numbers A000108.
a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(3^m)/n.
a(n) = (-1/2)^n * (1 - 3*Sum_{k=0..n-1} C(k)*(-6)^k), n >= 1, a(0) = 1, with C(n) = A000108(n) (Catalan).
a(n) = Sum_{k=0..n} A059365(n, k)*3^(n-k). - Philippe Deléham, Jan 19 2004
Given the semi-axes a,b of an ellipse, then Ramanujan gave the highly accurate formula for the perimeter p = Pi((a+b) + (3(a-b)^2)/(10(a+b) + sqrt(a^2 + 14ab + b^2))). If we let h = ((a-b)/(a+b))^2, then (p/(Pi(a+b))-1)/4 = (3/4)* h/(10 + sqrt(4 - 3*h)) = 1*(h/16) + 1*(h/16)^2 + 4*(h/16)^3 + 25*(h/16)^4 + ... . - Michael Somos, Apr 11 2007
G.f.: 1/(1-x/(1-3*x/(1-3*x/(1-3*x/(1-.... = 1/(1-x-3*x^2/(1-6*x-9*x^2/(1-6*x-9*x^2/(1-.... (continued fractions). - Paul Barry, Jan 30 2009
G.f.: 6/(5+sqrt(1-12*x)). - Harvey P. Dale, Mar 11 2011
From Gary W. Adamson, Jul 12 2011: (Start)
a(n) = upper left term in M^n, M = the infinite square production matrix:
  1, 1, 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: 2*n*a(n) + (-23*n+36)*a(n-1) + 6*(-2*n+3)*a(n-2) = 0. - R. J. Mathar, Dec 03 2012 (Formula verified and used for computations. - Fung Lam, Mar 05 2014)
a(n) ~ 3^(n+1) * 4^n / (25*n^(3/2)*sqrt(Pi)). - Vaclav Kotesovec, Mar 05 2014
a(n) = hypergeometric([1-n, n], [-n], 3) for n>0. - Peter Luschny, Nov 30 2014

A060757 a(n) = 4^(n^2).

Original entry on oeis.org

1, 4, 256, 262144, 4294967296, 1125899906842624, 4722366482869645213696, 316912650057057350374175801344, 340282366920938463463374607431768211456, 5846006549323611672814739330865132078623730171904, 1606938044258990275541962092341162602522202993782792835301376

Offset: 0

Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 23 2001

nonn

Number of n X n matrices over GF(4).

a(n) = k^(n^2) with k = 2, 3, 4,... counts n X n matrices over GF(k). - Vincenzo Origlio (vincenzo.origlio(AT)itc.cnr.it), Nov 14 2002

Harry J. Smith, Table of n, a(n) for n = 0..40

Cf. A060761.
Cf. A060722.
Cf. A118185.

f[n_]:=4^(n^2);f[Range[0,14]] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2011 *)

A060757(n):=4^(n^2)$
makelist(A060757(n),n,0,10); /* Martin Ettl, Nov 08 2012 */

a(n)={4^(n^2)} \\ Harry J. Smith, Jul 10 2009

def A060757(n): return 4^(n^2)
print([A060757(n) for n in range(0,11)]) # Stefano Spezia, Mar 28 2025

a(n) = A118185(2n,n). - Alois P. Heinz, Jun 29 2021

More terms from Philippe Deléham, Nov 19 2007

A155203 G.f.: A(x) = exp( Sum_{n>=1} 3^(n^2) * x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 3, 45, 6687, 10782369, 169490304819, 25016281429306077, 34185693516532070487615, 429210580094546346191627404353, 49269611092414945570325157106493868771

Offset: 0

Paul D. Hanna, Feb 04 2009

nonn

More generally, for m integer, exp( Sum_{n>=1} m^(n^2) * x^n/n ) is a power series in x with integer coefficients.

			G.f.: A(x) = 1 + 3*x + 45*x^2 + 6687*x^3 + 10782369*x^4 + 169490304819*x^5 +...
log(A(x)) = 3*x + 3^4*x^2/2 + 3^9*x^3/3 + 3^16*x^4/4 + 3^25*x^5/5 +...

Cf. A060722, A155204, A155205, A155206, A155812 (triangle), variants: A155200, A155207.

{a(n)=polcoeff(exp(sum(m=1,n+1,3^(m^2)*x^m/m)+x*O(x^n)),n)}

Equals column 0 of triangle A155812.
G.f. satisfies: A'(x)/A(x) = 3 + 27*x*A'(9*x)/A(9*x). - Paul D. Hanna, Nov 15 2022
a(n) ~ 3^(n^2)/n. - Vaclav Kotesovec, Oct 31 2024

A056989 Number of nonsingular n X n (-1,0,1)-matrices (over the reals).

Original entry on oeis.org

1, 2, 48, 11808, 27947520, 609653621760, 119288919620689920

Offset: 0

Eric W. Weisstein

It would be nice to have an estimate for the asymptotic rate of growth.

			a(1) = 2: [1], [ -1].
a(2) = 48: There are 8 choices for the first column, u (say) and then the 2nd column can be anything except 0, u, -u, so 6 choices, giving a total of 8*6 = 48.

Minfeng Wang, C++ program to calculate A056989
Eric Weisstein's World of Mathematics, Nonsingular Matrix
Index entries for sequences related to binary matrices

Cf. A055165, A053290, A056990, A055165, A002884, A046747, A057981, A060722.

(* A brute force solution up to n = 4 *) a[n_] := a[n] = (m = Array[x, {n, n}]; cnt = 0; iter = {#, -1, 1}& /@ Flatten[m]; Do[ If[ Det[m] != 0, cnt++], Evaluate[ Sequence @@ iter]]; cnt); Table[ Print[a[n]]; a[n], {n, 1, 4}] (* Jean-François Alcover, Oct 11 2012 *)

a(n) = A060722(n) - A057981(n). - Alois P. Heinz, Dec 02 2019

a(4) from Winston C. Yang (winston(AT)cs.wisc.edu), Aug 27 2000
Entry revised by N. J. A. Sloane, Jan 02 2007
a(5) from Giovanni Resta, Feb 20 2009
a(0)=1 prepended by Alois P. Heinz, Dec 02 2019
a(0)-a(5) confirmed and a(6) added by Minfeng Wang, May 01 2024

A057981 Number of singular n X n (-1,0,1)-matrices.

Original entry on oeis.org

0, 1, 33, 7875, 15099201, 237634987683, 30805715676309201

Offset: 0

Eric W. Weisstein, Oct 23 2000

Eric Weisstein's World of Mathematics, Singular Matrix.
Minfeng Wang, C++ program to calculate A057981

Complement of A056989.

Cf. A060722, A330134.

a(n) = A060722(n) - A056989(n). - Alois P. Heinz, Dec 02 2019

a(5) from Giovanni Resta, Feb 20 2009
a(0)=0 prepended by Alois P. Heinz, Dec 02 2019
a(0)-a(5) confirmed and a(6) added by Minfeng Wang, May 01 2024

A271570 Number of distinct eigenvalues of n X n matrices with elements {-1, 0, +1}.

Original entry on oeis.org

3, 21, 375, 24823

Offset: 1

Steven E. Thornton, Jul 13 2016

Steven E. Thornton & Robert M. Corless, The Bohemian Eigenvalue Project, Poster Presented at The International Symposium on Symbolic and Algebraic Computation (ISSAC 2016). Wilfrid Laurier University, July 19-22, 2016.

Robert Corless and Steven Thornton, The Bohemian Eigenvalue Project, 2017 poster.
Steven E. Thornton, GitHub repository for code used to generate this sequence.
Steven E. Thornton & Robert M. Corless, The Bohemian Eigenvalue Project.

Number of characteristic polynomials: A272658.

Cf. A060722.

(* Program not suitable to compute more than 3 terms *)
a[n_] := Module[{r, iter}, iter = Table[{r[k], {-1, 0, 1}}, {k, 1, n^2}]; Eigenvalues /@ (Table[Table[(r[# + j]& /@ Range[n]), {j, 0, n^2 - n, n}], Sequence @@ iter // Evaluate] // Flatten[#, n^2 - 1]&) // Flatten // Union // Length];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 3}] (* Jean-François Alcover, Jun 17 2018 *)

a(n) <= 3^(n^2). - Robert P. P. McKone, Sep 16 2023

A135397 a(n) = 2^n * 3^(n^2).

Original entry on oeis.org

1, 6, 324, 157464, 688747536, 27113235502176, 9606056659007943744, 30630314141519043787530624, 879023057994883196072409366855936, 227034361980435338213503810877004745115136

Offset: 0

Philippe Deléham, Dec 11 2007

Hankel transform of A110520.

G. C. Greubel, Table of n, a(n) for n = 0..25

A135397:=n->2^n*3^(n^2); seq(A135397(n), n=0..10); # Wesley Ivan Hurt, Nov 23 2013

Table[2^n*3^(n^2), {n,0,10}] (* Wesley Ivan Hurt, Nov 23 2013 *)

a(n)=2^n*3^n^2 \\ Charles R Greathouse IV, Oct 12 2016

a(n) = 2^n * 3^(n^2) = A000079(n) * A060722(n).

A271588 Number of matrices with multiple eigenvalues from the set of n X n matrices with elements {-1, 0, +1}.

Original entry on oeis.org

0, 19, 4629, 7171257, 89765448427

Offset: 1

Steven E. Thornton, Jul 13 2016

Steven E. Thornton and Robert M. Corless, The Bohemian Eigenvalue Project, Poster Presented at The International Symposium on Symbolic and Algebraic Computation (ISSAC 2016). Wilfrid Laurier University, July 19-22, 2016.

Robert Corless and Steven Thornton, The Bohemian Eigenvalue Project, 2017 poster.
GitHub repository for code used to generate this sequence.

Number of characteristic polynomials A272658.

Cf. A060722.

a[n_Integer?NonNegative] := a[n] = Module[{m, ei}, ei[matrix_] := Length[Select[Tally[Eigenvalues[matrix]], Last[#] > 1 &]] > 0; m = Tuples[Tuples[{-1, 0, 1}, n], n]; Count[m, mat_ /; ei[mat]]]; Table[a[i], {i, 1, 3}] (* Robert P. P. McKone, Sep 16 2023 *)

a(n) <= A060722(n) where A060722(n) = 3^(n^2); see Corless and Thornton poster link. Robert P. P. McKone, Sep 16 2023

A120840 5^(n^2)-3^(n^2).

Original entry on oeis.org

0, 2, 544, 1933442, 152544843904, 298022376588343682, 14551915078272216509641504, 17763568393763205317547489159863042, 542101086242748783319906107922486197863801344

Offset: 0

Mohammad K. Azarian, Aug 18 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..30

Cf. A007689, A001047.

[5^(n^2) - 3^(n^2): n in [0..10]]; // Vincenzo Librandi, Feb 27 2013

Table[5^(n^2) - 3^n^2, {n, 0, 10}] (* Vincenzo Librandi, Feb 27 2013 *)

a(n)=5^(n^2)-3^(n^2) \\ Charles R Greathouse IV, Feb 27 2013

a(n) = A060758(n)-A060722(n)=A095860(5,n)-A095860(3,n). - R. J. Mathar, Apr 24 2007

Corrected by Ray Chandler, Sep 06 2006

A133461 4^n*3^(n^2).

Original entry on oeis.org

1, 12, 1296, 1259712, 11019960576, 867623536069632, 614787626176508399616, 3920680210114437604803919872, 225029902846690098194536797915119616, 116241593333982893165313951169026429498949632

Offset: 0

Philippe Deléham, Nov 28 2007

Hankel transform of A039610.

a(n)=4^n*3^(n^2)=A000302(n)*A060722(n).

cp's OEIS Frontend

A064063 Generalized Catalan numbers C(3; n).

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A060757 a(n) = 4^(n^2).

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A155203 G.f.: A(x) = exp( Sum_{n>=1} 3^(n^2) * x^n/n ), a power series in x with integer coefficients.

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A056989 Number of nonsingular n X n (-1,0,1)-matrices (over the reals).

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A057981 Number of singular n X n (-1,0,1)-matrices.

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A271570 Number of distinct eigenvalues of n X n matrices with elements {-1, 0, +1}.

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A135397 a(n) = 2^n * 3^(n^2).

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