A098400 -id:A098400 - cp's OEIS frontend

A183772 T(n,k) = 1/32 the number of (n+1) X (k+1) binary arrays with equal numbers of 2 X 2 subblocks with sum mod two being 0 and 1.

0, 1, 1, 0, 6, 0, 12, 40, 40, 12, 0, 280, 0, 280, 0, 160, 2016, 7392, 7392, 2016, 160, 0, 14784, 0, 205920, 0, 14784, 0, 2240, 109824, 1555840, 5912192, 5912192, 1555840, 109824, 2240, 0, 823680, 0, 173065984, 0, 173065984, 0, 823680, 0, 32256, 6223360

Offset: 1

R. H. Hardin, Jan 06 2011

Table starts
.....0........1...........0..............12..................0
.....1........6..........40.............280...............2016
.....0.......40...........0............7392..................0
....12......280........7392..........205920............5912192
.....0.....2016...........0.........5912192..................0
...160....14784.....1555840.......173065984........19855042560
.....0...109824...........0......5134924800..................0
..2240...823680...346131968....153876579840.....70577422755840
.....0..6223360...........0...4646469273600..................0
.32256.47297536.79420170240.141154845511680.258888921984516096

			Some solutions for 4 X 3:
..0..1..0....1..1..0....1..0..0....0..1..1....1..1..1....0..0..0....0..0..1
..0..1..1....0..1..0....0..1..0....1..0..1....0..0..0....1..0..0....1..1..1
..0..1..1....1..1..0....1..0..0....1..0..0....1..0..1....1..0..1....0..0..1
..1..1..0....1..1..1....0..1..0....1..1..1....0..1..1....0..1..1....1..0..1

R. H. Hardin, Table of n, a(n) for n = 1..220

Column 1 is A098400(n/2-1).

Column 2 is A069720.

A098399 a(n) = 3^nbinomial(2n+1, n).

Original entry on oeis.org

1, 9, 90, 945, 10206, 112266, 1250964, 14073345, 159497910, 1818276174, 20827527084, 239516561466, 2763652632300, 31979409030900, 370961144758440, 4312423307816865, 50227047938102310, 585982225944526950, 6846739692614999100, 80106854403595489470, 938394580156404305220

Offset: 0

Paul Barry, Sep 06 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..900

Cf. A000108, A001700, A005159, A069720, A069723, A098400.

[3^n*Binomial(2*n+1, n): n in [ 0..20]]; // Vincenzo Librandi, Nov 24 2012

Z:=(1-sqrt(1-3*z))*4^n/sqrt(1-3*z)/6: Zser:=series(Z, z=0, 32): seq(coeff(Zser, z, n), n=1..18); # Zerinvary Lajos, Jan 01 2007

Table[3^n Binomial[2n+1,n], {n,0,20}] (* Harvey P. Dale, Mar 28 2012 *)

a(n)=binomial(2*n+1,n)*3^n \\ Charles R Greathouse IV, Oct 23 2023

[3^n*binomial(2*n+1, n) for n in range(21)] # G. C. Greubel, Dec 27 2023

G.f.: (1-sqrt(1-12*x))/(6*x*sqrt(1-12*x)).
E.g.f.: a(n) = n!* [x^n] exp(6*x)*(BesselI(0, 6*x) + BesselI(1, 6*x)). - Peter Luschny, Aug 25 2012
(n+1)*a(n) - 6*(2*n+1)*a(n-1) = 0. - R. J. Mathar, Nov 24 2012
From G. C. Greubel, Dec 27 2023: (Start)
a(n) = 3^n * (2*n+1)*A000108(n).
a(n) = (2*n+1)*A005159(n).
a(n) = 3^n * A001700(n). (End)
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=0} 1/a(n) = 6/11 + 72*arcsin(1/(2*sqrt(3)))/(11*sqrt(11)).
Sum_{n>=0} (-1)^n/a(n) = 6/13 + 72*arcsinh(1/(2*sqrt(3)))/(13*sqrt(13)). (End)

A098402 a(n) = (0^n + 4^n * binomial(2*n,n))/2.

Original entry on oeis.org

1, 4, 48, 640, 8960, 129024, 1892352, 28114944, 421724160, 6372720640, 96865353728, 1479398129664, 22684104654848, 348986225459200, 5384358907084800, 83278084429578240, 1290810308658462720, 20045524793284362240, 311819274562201190400, 4857816066863765913600

Offset: 0

Paul Barry, Sep 06 2004

It seems that a(n) is the number of pairs of binary vectors of length 2*n which are orthogonal. (Define binary vectors here to have components of value +1 or -1. There are no pairs of binary vectors of odd length which are orthogonal.) For example, the a(1) = 4 pairs of binary vectors of length 2 are (-1,-1) and (1,-1), (-1,-1) and (-1,1), (1,-1) and (1,1), (-1,1) and (1,1). Tested up to and including a(8). - R. J. Mathar, Apr 15 2013

Tested up to and including a(210). - R. H. Hardin, May 08 2013

G. C. Greubel, Table of n, a(n) for n = 0..825

Cf. A055372, A069723, A069720, A098400, A098401.

[(0^n + 4^n*(n+1)*Catalan(n))/2: n in [0..40]]; // G. C. Greubel, Dec 27 2023

Table[(Boole[n == 0] + 4^n Binomial[2 n, n])/2, {n, 0, 18}] (* or *)
CoefficientList[Series[8 x/(# (1 - #)) &@ Sqrt[1 - 16 x], {x, 0, 18}], x] (* Michael De Vlieger, Aug 03 2016 *)

[(4^n*binomial(2*n,n) + int(n==0))/2 for n in range(41)] # G. C. Greubel, Dec 27 2023

G.f.: 8*x/( sqrt(1 - 16*x)*(1 - sqrt(1 - 16*x)) ).
a(n+1) = 4*A098400(n).
n*a(n) - 8*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Nov 09 2012
a(n) ~ 16^n/(2*sqrt(Pi*n)). - Ilya Gutkovskiy, Aug 03 2016
a(n) = A055372(2*n,n). - Alois P. Heinz, Jan 21 2020
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=0} 1/a(n) = 17/15 + 32*arcsin(1/4)/(15*sqrt(15)).
Sum_{n>=0} (-1)^n/a(n) = 15/17 - 32*arcsinh(1/4)/(17*sqrt(17)). (End)

A098401 a(n) = (0^n + 3^nbinomial(2n,n))/2.

Original entry on oeis.org

1, 3, 27, 270, 2835, 30618, 336798, 3752892, 42220035, 478493730, 5454828522, 62482581252, 718549684398, 8290957896900, 95938227092700, 1112883434275320, 12937269923450595, 150681143814306930, 1757946677833580850, 20540219077844997300, 240320563210786468410

Offset: 0

Paul Barry, Sep 06 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..900

Cf. A069723, A069720, A098399, A098400, A098402.

[(0^n + 3^n * Binomial(2*n, n))/2: n in [ 0..20]]; // Vincenzo Librandi, Nov 24 2012

CoefficientList[Series[(6x)/(Sqrt[1-12x](1-Sqrt[1-12x])),{x,0,30}],x] (* Harvey P. Dale, Nov 29 2023 *)
Table[(3^n*Binomial[2*n,n] +Boole[n==0])/2, {n,0,40}] (* G. C. Greubel, Dec 27 2023 *)
a[n_] := 3^n*HypergeometricPFQ[{-n, -n + 1}, {1}, 1]; Flatten[Table[a[n], {n,0,20}]] (* Detlef Meya, May 21 2024 *)

[(3^n*binomial(2*n,n) + int(n==0))/2 for n in range(41)] # G. C. Greubel, Dec 27 2023

a(n+1) = 3*A098399(n).
G.f.: 6*x/(sqrt(1-12*x)*(1-sqrt(1-12*x))).
n*a(n) - 6*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Nov 24 2012
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=0} 1/a(n) = 13/11 + 24*arcsin(1/(2*sqrt(3)))/(11*sqrt(11)).
Sum_{n>=0} (-1)^n/a(n) = 11/13 - 24*arcsinh(1/(2*sqrt(3)))/(13*sqrt(13)). (End)
a(n) = 3^n*hypergeom([-n, -n + 1], [1], 1). - Detlef Meya, May 21 2024

A188632 Size of optimal DNA code of length n and minimal distance 1 over alphabet of size 4.

Original entry on oeis.org

1, 4, 12, 44, 160, 640, 2240, 8912, 32256

Offset: 1

N. J. A. Sloane, Apr 05 2011

nonn

See Sun et al. reference for precise definition.

J. Sun, S. Houghten and J. Ross, Edit metric codes with combinatorial DNA constraints, Congessus Numerant., 204 (2010), 65-92 (see Table 1).

One bisection appears to be A098400.

cp's OEIS Frontend

A183772 T(n,k) = 1/32 the number of (n+1) X (k+1) binary arrays with equal numbers of 2 X 2 subblocks with sum mod two being 0 and 1.

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A098399 a(n) = 3^n*binomial(2*n+1, n).

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A098402 a(n) = (0^n + 4^n * binomial(2*n,n))/2.

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A098401 a(n) = (0^n + 3^n*binomial(2*n,n))/2.

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A188632 Size of optimal DNA code of length n and minimal distance 1 over alphabet of size 4.

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Formula

A098399 a(n) = 3^nbinomial(2n+1, n).

A098401 a(n) = (0^n + 3^nbinomial(2n,n))/2.