A098402 -id:A098402 - cp's OEIS frontend

A225987 T(n,k)=Number of pairs of orthogonal (-x,y) vectors of length k*(x+y), where x/y is the n-th rational <= 1, ordered first by y and then x, e.g. 1/1, 1/2, 1/3, 2/3, 1/4, 3/4 ...

4, 48, 9, 640, 390, 20, 8960, 20160, 3192, 60, 129024, 1106424, 652784, 48720, 45, 1892352, 62606544, 134432480, 41801760, 26010, 420, 28114944, 3611217792, 28244153600, 36951200000, 21594300, 8168160, 102, 421724160, 211096262400

Offset: 1

R. H. Hardin, May 22 2013

Table starts
..4....48......640........8960.........129024...........1892352
..9...390....20160.....1106424.......62606544........3611217792
.20..3192...652784...134432480....28244153600.....6052498776192
.60.48720.41801760.36951200000.33792065269760.31520952625612800
.45.26010.21594300.15593341800.11432293516320..8690522384325600

			Some solutions for n=1 k=4
.-1..1....1..1...-1.-1...-1.-1...-1..1....1..1...-1..1...-1.-1...-1.-1...-1..1
..1..1...-1.-1...-1..1...-1..1....1..1...-1..1...-1.-1...-1..1...-1..1...-1.-1
.-1..1...-1..1....1..1...-1.-1...-1..1....1.-1....1.-1...-1.-1....1.-1....1.-1
.-1.-1...-1.-1....1.-1...-1.-1....1..1...-1.-1....1.-1...-1..1....1.-1....1.-1
..1.-1....1.-1...-1.-1....1.-1...-1.-1...-1.-1....1..1...-1..1...-1..1...-1.-1
..1.-1...-1..1....1..1....1..1...-1..1....1.-1....1.-1...-1.-1...-1.-1...-1.-1
..1..1....1..1...-1..1....1.-1....1.-1....1.-1....1..1....1..1...-1.-1...-1.-1
..1..1....1.-1...-1..1...-1..1...-1.-1...-1.-1....1..1....1.-1...-1.-1....1.-1

R. H. Hardin, Table of n, a(n) for n = 1..1056

Row 1 is A098402.

A055372 Invert transform of Pascal's triangle A007318.

Original entry on oeis.org

1, 1, 1, 2, 4, 2, 4, 12, 12, 4, 8, 32, 48, 32, 8, 16, 80, 160, 160, 80, 16, 32, 192, 480, 640, 480, 192, 32, 64, 448, 1344, 2240, 2240, 1344, 448, 64, 128, 1024, 3584, 7168, 8960, 7168, 3584, 1024, 128, 256, 2304, 9216, 21504, 32256, 32256, 21504, 9216, 2304, 256

Offset: 0

Christian G. Bower, May 16 2000

Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 10 2005

T(n,k) is the number of nonempty bit strings with n bits and exactly k 1's over all strings in the sequence. For example, T(2,1)=4 because we have {(01)},{(10)},{(0),(1)},{(1),(0)}. - Geoffrey Critzer, Apr 06 2013

			Triangle begins:
  1;
  1,  1;
  2,  4,  2;
  4, 12, 12,  4;
  8, 32, 48, 32,  8;
  ...

N. J. A. Sloane, Transforms
Index entries for triangles and arrays related to Pascal's triangle

Row sums give A081294. Cf. A000079, A007318, A055373, A055374.
Cf. A134309.
T(2n,n) gives A098402.

nn=10;f[list_]:=Select[list,#>0&];a=(x+y x)/(1-(x+y x));Map[f,CoefficientList[Series[1/(1-a),{x,0,nn}],{x,y}]]//Grid  (* Geoffrey Critzer, Apr 06 2013 *)

a(n,k) = 2^(n-1)*C(n, k), for n>0.
G.f.: A(x, y)=(1-x-xy)/(1-2x-2xy).
As an infinite lower triangular matrix, equals A134309 * A007318. - Gary W. Adamson, Oct 19 2007
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A011782(n), A081294(n), A081341(n), A092811(n), A093143(n), A067419(n) for x = -1, 0, 1, 2, 3, 4, 5 respectively. - Philippe Deléham, Feb 05 2012

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

cp's OEIS Frontend

A225987 T(n,k)=Number of pairs of orthogonal (-x,y) vectors of length k*(x+y), where x/y is the n-th rational <= 1, ordered first by y and then x, e.g. 1/1, 1/2, 1/3, 2/3, 1/4, 3/4 ...

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A055372 Invert transform of Pascal's triangle A007318.

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

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cp's OEIS Frontend

A225987 T(n,k)=Number of pairs of orthogonal (-x,y) vectors of length k*(x+y), where x/y is the n-th rational <= 1, ordered first by y and then x, e.g. 1/1, 1/2, 1/3, 2/3, 1/4, 3/4 ...

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A055372 Invert transform of Pascal's triangle A007318.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

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