login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A107267 A square array of Motzkin related transforms, read by antidiagonals. 10
1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 4, 6, 3, 1, 0, 9, 20, 12, 4, 1, 0, 21, 72, 54, 20, 5, 1, 0, 51, 272, 261, 112, 30, 6, 1, 0, 127, 1064, 1323, 672, 200, 42, 7, 1, 0, 323, 4272, 6939, 4224, 1425, 324, 56, 8, 1, 0, 835, 17504, 37341, 27456, 10625, 2664, 490, 72, 9, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Rows are transforms of k^n, k>=0, under the matrix A107131. As a number triangle, with T(n,k)=if(k<=n,sum{j=0..n-k, (1/(j+1))C(j+1,n-k-j+1)C(n-k,j)k^j},0), row sums are A107268 and diagonal sums are A107269. Rows are series reversions of x/(1+kx+kx^2), k>=0. Conjecture: rows count weighted Motzkin paths.

Row k counts colored Motzkin paths, where H(1,0) and U(1,1) each have k colors and D(1,-1) one color. - Paul Barry, May 16 2005

LINKS

Seiichi Manyama, Antidiagonals n = 0..139, flattened

FORMULA

Number array T(n,k) = Sum_{j=0..k} n^j * binomial(k,j) * binomial(j+1,k-j+1)/(j+1).

G.f. of row k: 1/(1 - k*x - k*x^2/(1 - k*x - k*x^2/(1 - k*x - k*x^2/(1 - k*x - k*x^2/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Sep 21 2017

From Seiichi Manyama, May 05 2019: (Start)

T(n,k) = Sum_{j=0..floor(k/2)} n^(k-j) * binomial(k,2*j) * binomial(2*j,j)/(j+1) = Sum_{j=0..floor(k/2)} n^(k-j) * binomial(k,2*j) * A000108(j).

(k+2) * T(n,k) = n * (2*k+1) * T(n,k-1) - n * (n-4) * (k-1) * T(n,k-2). (End)

EXAMPLE

Array begins

  1, 0,  0,   0,    0,     0,      0, ...

  1, 1,  2,   4,    9,    21,     51, ...

  1, 2,  6,  20,   72,   272,   1064, ...

  1, 3, 12,  54,  261,  1323,   6939, ...

  1, 4, 20, 112,  672,  4224,  27456, ...

  1, 5, 30, 200, 1425, 10625,  81875, ...

  1, 6, 42, 324, 2664, 22896, 203256, ...

CROSSREFS

Rows n=0..6 give A000007, A001006, A071356, A107264, A003645, A107265, A107266.

Main diagonal gives A292716.

Cf. A000108.

Sequence in context: A191348 A198792 A196182 * A320530 A347986 A191239

Adjacent sequences:  A107264 A107265 A107266 * A107268 A107269 A107270

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, May 15 2005

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 17 13:24 EDT 2022. Contains 353746 sequences. (Running on oeis4.)