cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A060656 a(n) = 2*a(n-1)*a(n-2)/a(n-3), with a(0)=a(1)=1.

Original entry on oeis.org

1, 1, 2, 4, 16, 64, 512, 4096, 65536, 1048576, 33554432, 1073741824, 68719476736, 4398046511104, 562949953421312, 72057594037927936, 18446744073709551616, 4722366482869645213696, 2417851639229258349412352
Offset: 0

Views

Author

Henry Bottomley, Apr 18 2001

Keywords

Comments

a(n+1) is the Hankel transform of A135052. - Paul Barry, Nov 15 2007
a(n+1) is the Hankel transform of the aerated large Schroeder numbers. a(n) and a(n+1) both satisfy the trivial Somos-4 recurrence u(n)=4*u(n-2)^2/u(n-4). Associated with the elliptic curve y^2=1-6x^2+x^4 via Schroeder numbers. - Paul Barry, Dec 08 2009
Hankel transform of A089324. - Paul Barry, Mar 01 2010
a(n+1) is the number of n X n binary matrices that are symmetric about both diagonals (bisymmetric). For the derivation of this result, see the link below. - Dennis P. Walsh, Apr 03 2014
1 followed by {a(n-1)}A078495).%20-%20_Vladimir%20Shevelev">(n>=1) is the Somos-3 sequence: b(0)=b(1)=b(2)=1;for n>=3, b(n)=2*b(n-1)*b(n-2)/b(n-3) (cf. comment in A078495). - _Vladimir Shevelev, Apr 20 2016
If the Hankel transform is defined as in the link 'Sequence transformations' then a(n) is the Hankel transform of A151374. - Peter Luschny, Nov 30 2016

Examples

			a(6) = 2*64*16/4 = 512.
G.f. = 1 + x + 2*x^2 + 4*x^3 + 16*x^4 + 64*x^5 + 512*x^6 + 4096*x^7 + ...

Links

Crossrefs

Cf. A002416, A002620, A016116, A038754, A089324, A135052, A262666, A078495, A151374.

Programs

  • Maple
    A060656:=n->2^floor(n^2/4); seq(A060656(n), n=0..20); # Wesley Ivan Hurt, Apr 30 2014
  • Mathematica
    a[ n_] := 2^Quotient[n^2, 4]; (* Michael Somos, Jan 24 2014 *)
nxt[{a_,b_,c_}]:={b,c,(2c*b)/a}; NestList[nxt,{1,1,2},20][[All,1]] (* Harvey P. Dale, Nov 26 2017 *)
  • PARI
    { for (n=0, 100, write("b060656.txt", n, " ", 2^(n^2\4)); ) } \\ Harry J. Smith, Jul 09 2009
  • PARI
    {a(n) = 2^(n^2\4)}; /* Michael Somos, Jan 24 2014 */

Formula

a(n) = 2^floor( n^2/4 ) = a(n - 1) * 2^floor( n/2 ) = a(n - 2) * 2^(n - 1) = a(n - 1) * A016116(n) = 2^A002620(n).
0 = a(n) * a(n+3) + a(n+1) * ( -2*a(n+2) ) for all n in Z. - Michael Somos, Jan 24 2014
0 = a(n) * a(n+4) + a(n+2) * ( -4*a(n+2) ) for all n in Z. - Michael Somos, Jan 24 2014

A184948 Triangle read by rows: SM(n,m) is the number of symmetric 0-1 matrices of order n such that the total number of 1's is m (n >= 1, 0 <= m <= n^2).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 1, 1, 3, 6, 10, 12, 12, 10, 6, 3, 1, 1, 4, 12, 28, 52, 84, 116, 140, 150, 140, 116, 84, 52, 28, 12, 4, 1, 1, 5, 20, 60, 150, 326, 620, 1060, 1635, 2295, 2952, 3480, 3780, 3780, 3480, 2952, 2295, 1635, 1060, 620, 326, 150, 60, 20, 5, 1
Offset: 1

Views

Author

N. J. A. Sloane, Feb 03 2011, based on a posting to the Sequence Fans Mailing List by Brendan McKay, Feb 02 2011

Keywords

Examples

			Triangle begins:
SM(1, m) = 1, 1
SM(2, m) = 1, 2, 2, 2, 1
SM(3, m) = 1, 3, 6, 10, 12, 12, 10, 6, 3, 1
SM(4, m) = 1, 4, 12, 28, 52, 84, 116, 140, 150, 140, 116, 84, 52, 28, 12, 4, 1
SM(5, m) = 1, 5, 20, 60, 150, 326, 620, 1060, 1635, 2295, 2952, 3480, 3780, 3780, 3480, 2952, 2295, 1635, 1060, 620, 326, 150, 60, 20, 5, 1
...

Links

Crossrefs

Row sums give A006125(n+1).
Cf. A262666.

Programs

  • Mathematica
    row[n_] := CoefficientList[(1+x)^n (1+x^2)^(n(n-1)/2), x];
Array[row, 5] // Flatten (* Jean-François Alcover, Mar 19 2019 *)
  • PARI
    {SM(n,k)=polcoeff((1+x^2)^(n*(n-1)/2)*(1+x)^n,k)} \\ Paul D. Hanna
  • PARI
    {SM(n,k)=local(A); A=sum(m=1, n, x^m*(1+y)^m*prod(k=1, m, (1-x*(1+y)*(1+y^2)^(2*k-2))/(1-x*(1+y)*(1+y^2)^(2*k-1)+x*O(x^n))));polcoeff(polcoeff(A, n,x),k,y)} \\ Paul D. Hanna

Formula

SM(n,m) is the sum of binomial(n,k) * binomial(n*(n-1)/2,(m-k)/2) over those k with the same parity as m. To see this consider that k is the number of 1s on the diagonal.
From Robert Israel, Feb 02 2011: (Start)
According to Maple,
> simplify(sum(binomial(n,2*j)*binomial(r,M-j),j=0..M)) assuming posint;
binomial(r,M)*hypergeom([-M, -1/2*n, 1/2-1/2*n],[1/2, r-M+1],-1)
> simplify(sum(binomial(n,2*k+1)*binomial(r,M-k),k=0..M)) assuming posint;
n*binomial(r,M)*hypergeom([-M, 1-1/2*n, 1/2-1/2*n],[3/2, r-M+1],-1)
If m is even you want the first formula with r=n*(n-1)/2 and M=m/2.
If m is odd the second formula with r=n*(n-1)/2 and M=(m+1)/2.
Thus for n=5 and m=6,
binomial(10,3)*hypergeom([-3,-5/2,-2],[1/2,8],-1) = 620
and for n=5 and m=5,
5*binomial(10,3)*hypergeom([-3, -3/2, -2],[3/2, 8],-1) = 1060. (End)
G.f. for row n: (1+x)^n*(1+x^2)^(n*(n-1)/2) for n>=1. - Paul D. Hanna, Feb 03 2011
G.f.: A(x,y) = Sum_{n>=1} x^n*(1+y)^n*Product_{k=1..n} (1-x(1+y)(1+y^2)^(2k-2))/(1-x(1+y)(1+y^2)^(2k-1)) due to a q-series identity. - Paul D. Hanna, Feb 03 2011
Sum_{k>=0..n^2} k*SM(n,k) = n^2/2 * 2^(n(n+1)/2).
SM(n,m) = Sum_{k=0..floor(m/2)} C(C(n,2),k)*C(n,m-2*k), from equation (11) in the Cameron et al., reference. - L. Edson Jeffery, Feb 29 2012

A262742 Irregular table read by rows: T(n,k) is the number of binary symmetric n X n matrices with exactly k 1's; n>=0, 0<=k<=n^2. Where the symmetry axes are in horizontal and vertical.

Original entry on oeis.org

1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 0, 0, 0, 4, 0, 0, 0, 6, 0, 0, 0, 4, 0, 0, 0, 1, 1, 1, 4, 4, 10, 10, 20, 20, 31, 31, 40, 40, 44, 44, 40, 40, 31, 31, 20, 20, 10, 10, 4, 4, 1, 1, 1, 0, 0, 0, 9, 0, 0, 0, 36, 0, 0, 0, 84, 0, 0, 0, 126, 0, 0
Offset: 0

Views

Author

Kival Ngaokrajang, Sep 29 2015

Keywords

Comments

The row length of this irregular triangle is n^2+1 = A002522(n).
Inspired by A262666, but rotating the diagonal and antidiagonal symmetry axis to horizontal and vertical axes.
From Wolfdieter Lang, Oct 12 2015 (Start):
Double symmetry of n X n matrix M: M(i, j) = M(n-i+1, j) = M(i, n-j+1) (= M(n-i+1, n-j+1)), here with entries from {0, 1}.
Due to 0 <-> 1 flip the rows are symmetric.
The number of independent entries in such an n X n doubly symmetric matrix is A008794(n+1) (squares repeated). Therefore, the row sums give repeated A002416 (omitting the first 1): 1, 2, 2, 16, 16, 512, 512, ... (End) - Wolfdieter Lang, Oct 12 2015

Examples

			Irregular table begins:
n\k 0   1   2   3   4   5   6   7   8   9   ...
0:  1
1:  1   1
2:  1   0   0   0   1
3:  1   1   2   2   2   2   2   2   1   1
...
Row 4: 1, 0, 0, 0, 4, 0, 0, 0, 6, 0, 0, 0, 4, 0, 0, 0, 1;
Row 5: 1, 1, 4, 4, 10, 10, 20, 20, 31, 31, 40, 40, 44, 44, 40, 40, 31, 31, 20, 20, 10, 10, 4, 4, 1, 1.
...

Links

Crossrefs

Cf. A262666,A002522, A008794, A002416.

Extensions

More terms from Alois P. Heinz, Sep 29 2015
Showing 1-3 of 3 results.

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