A060757 -id:A060757 - cp's OEIS frontend

A118185 Triangle T(n,k) = 4^(k*(n-k)) for n>=k>=0, read by rows.

1, 1, 1, 1, 4, 1, 1, 16, 16, 1, 1, 64, 256, 64, 1, 1, 256, 4096, 4096, 256, 1, 1, 1024, 65536, 262144, 65536, 1024, 1, 1, 4096, 1048576, 16777216, 16777216, 1048576, 4096, 1, 1, 16384, 16777216, 1073741824, 4294967296, 1073741824, 16777216, 16384, 1

Offset: 0

Paul D. Hanna, Apr 15 2006

For any column vector C, the matrix product of T*C transforms the g.f. of C: Sum_{n>=0} c(n)*x^n into the g.f.: Sum_{n>=0} c(n)*x^n/(1-4^n*x).

Matrix power T^m satisfies: [T^m](n,k) = [T^m](n-k,0)*T(n,k) for all m and so the triangle has an invariant character. For example, the matrix inverse is defined by [T^-1](n,k) = A118188(n-k)*T(n,k); also, the matrix log is given by [log(T)](n,k) = A118189(n-k)*T(n,k).

			A(x,y) = 1/(1-xy) + x/(1-4xy) + x^2/(1-16xy) + x^3/(1-64xy) + ...
Triangle begins:
  1;
  1,    1;
  1,    4,       1;
  1,   16,      16,        1;
  1,   64,     256,       64,        1;
  1,  256,    4096,     4096,      256,       1;
  1, 1024,   65536,   262144,    65536,    1024,    1;
  1, 4096, 1048576, 16777216, 16777216, 1048576, 4096, 1; ...
The matrix inverse T^-1 starts:
        1;
       -1,      1;
        3,     -4,       1;
      -33,     48,     -16,     1;
     1407,  -2112,     768,   -64,    1;
  -237057, 360192, -135168, 12288, -256, 1; ...
where [T^-1](n,k) = A118188(n-k)*4^(k*(n-k)).

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A118186 (row sums), A118187 (antidiagonal sums), A118188, A118189.
Cf. A117401 (m=0), A118180 (m=1), this sequence (m=2), A118190 (m=3), A158116 (m=4), A176642 (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15).
T(2n,n) gives A060757.

[4^(k*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 29 2021

Table[4^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 29 2021 *)

PARI
```
T(n, k)=if(n
				
```

flatten([[4^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 29 2021

G.f.: A(x,y) = Sum_{n>=0} x^n/(1-4^n*x*y).
G.f. satisfies: A(x,y) = 1/(1-x*y) + x*A(x,4*y).
T(n,k) = (1/n)*( 4^(n-k)*k*T(n-1,k-1) + 4^k*(n-k)*T(n-1,k) ), where T(i,j)=0 if j>i. - Tom Edgar, Feb 20 2014
T(n, k, m) = (m+2)^(k*(n-k)) with m = 2. - G. C. Greubel, Jun 29 2021

A285388 a(n) = numerator of ((1/n) * Sum_{k=0..n^2-1} binomial(2k,k)/4^k).

Original entry on oeis.org

1, 35, 36465, 300540195, 79006629023595, 331884405207627584403, 22292910726608249789889125025, 11975573020964041433067793888190275875, 411646257111422564507234009694940786177843149765, 56592821660064550728377610673427602421565368547133335525825

Offset: 1

Ralf Steiner, Apr 18 2017

Editorial comment: This sequence arose from Ralf Steiner's attempt to prove Legendre's conjecture that there is a prime between N^2 and (N+1)^2 for all N. - N. J. A. Sloane, May 01 2017

Indranil Ghosh, Table of n, a(n) for n = 1..40

Cf. A000079, A000265, A056220, A060757, A201555, A285389 (denominators), A285406, A280655 (similar), A190732 (2/sqrt(Pi)), A285738 (greatest prime factor), A285717, A285730, A285786, A286264, A000290 (n^2), A056220 (2*n^2 -1), A286127 (sum a(n-1)/a(n)).

[Numerator( n*(n^2+1)*Catalan(n^2)/2^(2*n^2-1) ): n in [1..21]]; // G. C. Greubel, Dec 11 2021

Table[Numerator[Sum[Binomial[2k,k]/4^k,{k,0,n^2-1}]/n],{n,1,10}]
Numerator[Table[2^(1-2 n^2) n Binomial[2 n^2,n^2],{n,1,10}]] (* Ralf Steiner, Apr 22 2017 *)

A285388(n) = numerator((2^(1 - 2*(n^2)))*n*binomial(2*(n^2), n^2)); \\ Antti Karttunen, Apr 27 2017

a(n) = m=n*binomial(2*n^2, n^2);m>>valuation(m,2) \\ David A. Corneth, Apr 27 2017

from sympy import binomial, Integer
def a(n): return (Integer(2)**(1 - 2*n**2)*n*binomial(2*n**2, n**2)).numerator # Indranil Ghosh, Apr 27 2017

[numerator( n*(n^2+1)*catalan_number(n^2)/2^(2*n^2-1) ) for n in (1..20)] # G. C. Greubel, Dec 11 2021

a(n) is numerator of n*binomial(2 n^2, n^2)/2^(2*n^2 - 1). - Ralf Steiner, Apr 26 2017
a(n) = numerator(n*A201555(n) / (A060757(n)/2)) = n*A201555(n) / 2^(A285717(n)) = A000265(n*A201555(n)). [Using Ralf Steiner's formula and A285717(n) <= A056220(n), cf. A285406.] - Antti Karttunen, Apr 27 2017
Limit_{i->oo} a(i)*A285389(i+1)/(a(i+1)*A285389(i)) = 1. - Ralf Steiner, May 03 2017

Edited (including the removal of the author's claim that this leads to a proof of the Legendre conjecture) by N. J. A. Sloane, May 01 2017
Formula section edited by M. F. Hasler, May 02 2017
Edited by N. J. A. Sloane, May 10 2017

A302174 Smallest solution x of x^n + y^(n+1) = z^(n+2), x, y, z >= 1.

Original entry on oeis.org

1, 2, 27, 256, 472392, 262144, 13759414272, 4294967296, 4057816381784064

Offset: 0

Jacques Tramu, Apr 07 2018

From M. F. Hasler, Apr 13 2018: (Start)
Proofs for the upper limits given in the formula section:
For odd n = 2k-1, x = 2^(2*k^2) yields a solution with x^(2k-1) = y^(2k) = 1/2*z^(2k+1), y = 2^(k(2k-1)) and z = 2^(2k(k-1)+1), because 2*k^2*(2k-1) + 1 = (2k+1)*(2k(k-1)+1).
For even n = 2k, x = 2^(k*(2k+1))*3^(2k+2) yields a solution, with y = 2^(2*k^2)*3^(2k+1) and z = 2^(2*k^2-k+1)*3^(2k), because for the exponents of 3, (2k+1)^2 = (2k+2)2k + 1 and the factor 1+3 = 2^2 adds 2 to the (identical) exponent of 2 in x^(2k) and y^(2k+1), to factor as 2*k^2(2k+1) + 2 = (2k+2)(k(2k-1)+1). (End)

			1^0 + 3^1 = 2^2, therefore a(0) = 1.
2^1 + 5^2 = 3^3, so a(1) = 2. (No solution can have x = 1 because z^3 - 1 = (z - 1)(z^2 + z + 1) cannot be a square: if a = z - 1, then z^2 + z + 1 = a^2 + 3a + 3 is congruent to 3 modulo any factor of a, and a = 3b yields z^3 - 1 = 9*b*(3*b^2 + 3b + 1), the last factor being congruent to 1 modulo any factor of b, and cannot be a square.)
27^2 + 18^3 = 9^4, so a(2) = 27.
256^3 + 64^4 = 32^5, so a(3) = 256.
472392^4 + 52488^5 = 8748^6, so a(4) = 472392.

Cf. A001105 (2*k^2), A060757 (4^k^2 = 2^(2k^2)), A000244 (3^k).
Conjectured to be a subsequence of A003586 (2^i*3^j).
Cf. A300564, A300565, A300566, A300567, A300568 (z^4 = x^2 + y^3, ..., z^8 = x^6 + y^7).

For odd n, a(n) <= 2^((n+1)^2/2); for even n, a(n) <= 2^(n*(n+1)/2)*3^(n+2).

We may conjecture that, for n > 4, a(n) is given by these upper limits.

Extended to a(0) = 1 by M. F. Hasler, Apr 13 2018

A076782 a(n) = 10^(n^2).

Original entry on oeis.org

1, 10, 10000, 1000000000, 10000000000000000, 10000000000000000000000000, 1000000000000000000000000000000000000, 10000000000000000000000000000000000000000000000000, 10000000000000000000000000000000000000000000000000000000000000000, 1000000000000000000000000000000000000000000000000000000000000000000000000000000000

Offset: 0

Vincenzo Origlio (vincenzo.origlio(AT)itc.cnr.it), Nov 15 2002

nonn

Number of n X n matrices over an alphabet of size 10.
The 10th term of this sequence is the googol (10^100).
The term in position 10^50 (sqrt(googol)) is the googolplex (10^googol).
a(n) = k^(n^2) with k = 2, 3, 4, ... counts n X n matrices over an alphabet of size k.

Vincenzo Librandi, Table of n, a(n) for n = 0..20
Eric Weisstein's World of Mathematics, Googol, Googolplex

Cf. A060757, A060758, A060759, A060760, A060761.

[10^(n^2): n in [0..10]]; // Vincenzo Librandi, May 30 2011

10^Range[0,10]^2 (* Vladimir Joseph Stephan Orlovsky, Feb 01 2012 *)

A076782(n):=10^(n^2)$
makelist(A076782(n),n,0,10); /* Martin Ettl, Nov 08 2012 */

A155207 G.f.: A(x) = exp( Sum_{n>=1} 4^(n^2) * x^n/n ), a power series in x with integer coefficients.

Original entry on oeis.org

1, 4, 136, 87904, 1074100576, 225184288253824, 787061981348092400896, 45273238870711805132010916864, 42535296046210357883346895894694749696, 649556283428320264374891976653586736162144180224

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 + 4*x + 136*x^2 + 87904*x^3 + 1074100576*x^4 +...
log(A(x)) = 4*x + 4^4*x^2/2 + 4^9*x^3/3 + 4^16*x^4/4 + 4^25*x^5/5 +...

Cf. A060757, A155208, A155209, A155210, variants: A155200, A155203.

{a(n)=polcoeff(exp(sum(m=1,n+1,4^(m^2)*x^m/m)+x*O(x^n)),n)}

G.f. satisfies: A'(x)/A(x) = 4 + 64*x*A'(16*x)/A(16*x). - Paul D. Hanna, Nov 15 2022

a(n) ~ 4^(n^2)/n. - Vaclav Kotesovec, Oct 31 2024

A076781 a(n) = 6^(n^2).

Original entry on oeis.org

1, 6, 1296, 10077696, 2821109907456, 28430288029929701376, 10314424798490535546171949056, 134713546244127343440523266742756048896, 63340286662973277706162286946811886609896461828096

Offset: 0

Vincenzo Origlio (vincenzo.origlio(AT)itc.cnr.it), Nov 15 2002

nonn

Number of n X n matrices over an alphabet of size 6.

a(n) = k^(n^2) with k = 2, 3, 4, ... counts n X n matrices over Z/kZ.

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

Cf. A060757, A060758, A060759, A060760, A060761.

[ 6^(n^2): n in [0..10]]; // Vincenzo Librandi, May 30 2011

6^Range[0,12]^2 (* Vladimir Joseph Stephan Orlovsky, Feb 01 2012 *)

More terms from Philippe Deléham, Nov 24 2007

a(5) corrected by Vincenzo Librandi, May 30 2011

A076783 a(n) = 11^(n^2).

Original entry on oeis.org

1, 11, 14641, 2357947691, 45949729863572161, 108347059433883722041830251, 30912680532870672635673352936887453361, 1067189571633593786424240872639621090354383081702091, 4457915684525902395869512133369841539490161434991526715513934826241

Offset: 0

Vincenzo Origlio (vincenzo.origlio(AT)itc.cnr.it), Nov 15 2002

Number of n X n matrices over GF(11).

a(n) = k^(n^2) with k = 2, 3, 4, ... counts n X n matrices over GF(k).

Vincenzo Librandi, Table of n, a(n) for n = 0..20

Cf. A060757, A060758, A060759, A060760, A060761.

[11^(n^2): n in [0..10]]; // Vincenzo Librandi, May 30 2011

11^Range[0,10]^2 (* Vladimir Joseph Stephan Orlovsky, Feb 01 2012 *)

More terms from Rick L. Shepherd, May 06 2008

A135292 a(n) = 5^n * 4^(n^2).

Original entry on oeis.org

1, 20, 6400, 32768000, 2684354560000, 3518437208883200000, 73786976294838206464000000, 24758800785707605497982484480000000, 132922799578491587290380706028034457600000000

Offset: 0

Philippe Deléham, Dec 04 2007

Hankel transform of A130976 .

G. C. Greubel, Table of n, a(n) for n = 0..40

Table[5^n*4^(n^2), {n, 0, 10}] (* G. C. Greubel, Oct 09 2016 *)

a(n)=5^n*4^(n^2) \\ Charles R Greathouse IV, Oct 09 2016

a(n)=5^n*4^(n^2)=A000351(n)*A060757(n).

A135398 a(n) = 3^n * 4^(n^2).

Original entry on oeis.org

1, 12, 2304, 7077888, 347892350976, 273593677362757632, 3442605166011971360784384, 693087965674784425268322477539328, 2232592609368277258783200799359831235362816

Offset: 0

Philippe Deléham, Dec 11 2007

Hankel transform of A132863 .

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

Table[3^n*4^(n^2), {n, 0, 10}] (* G. C. Greubel, Oct 12 2016 *)

a(n)=3^n*4^n^2 \\ Charles R Greathouse IV, Oct 12 2016

a(n) = 3^n*4^(n^2) = A000244(n)*A060757(n).

A377881 Number of ordered pairs of real n X n (0,1)-matrices that satisfy the equation A + B = A * B.

Original entry on oeis.org

1, 1, 2, 72, 3760, 210042

Offset: 0

Stuart E Anderson, Nov 10 2024

Matrix multiplication of A and B is commutative here.

If A + B = A * B then (A - I)*(B - I) = I, where I is the identity matrix. For integer matrices, the determinant of (A-I) must be +-1 and its inverse gives B-I. - Andrew Howroyd, Nov 12 2024

			One of the 72 solutions in 3x3 (1,0) matrices:
  A = {{0,0,0},{0,1,1},{1,1,1}},
  B = {{0,0,0},{1,1,1},{0,1,1}}
  A + B = {{0,0,0},{1,2,2},{1,2,2}}
  A * B = {{0,0,0},{1,2,2},{1,2,2}}

Stuart E Anderson, C++ program for NxN solutions
Math Stackexchange, If A+B=AB, A,B commute

Cf. A060757.

\\ See comments. Uses Gray code to generate A-I (called A here).
a(n)= { my(Id=matid(n), A=-Id); sum(f=0, 2^(n^2)-1, if(f, my(t=valuation(f,2), i=t\n+1, j=t%n+1); A[i,j]=if(i==j,-1,1)-A[i,j]); if(abs(matdet(A))==1, my(B=A^(-1)+Id); vecmin(B)>=0 && vecmax(B)<=1 && denominator(B)==1)) } \\ Andrew Howroyd, Nov 12 2024

a(4) corrected and a(5) from Andrew Howroyd, Nov 12 2024

cp's OEIS Frontend

A118185 Triangle T(n,k) = 4^(k*(n-k)) for n>=k>=0, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A285388 a(n) = numerator of ((1/n) * Sum_{k=0..n^2-1} binomial(2k,k)/4^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Python

Sage

Formula

Extensions

A302174 Smallest solution x of x^n + y^(n+1) = z^(n+2), x, y, z >= 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A076782 a(n) = 10^(n^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

A155207 G.f.: A(x) = exp( Sum_{n>=1} 4^(n^2) * x^n/n ), a power series in x with integer coefficients.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A076781 a(n) = 6^(n^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Extensions

A076783 a(n) = 11^(n^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs