login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A108553 Square array, read by antidiagonals, where row n equals the crystal ball sequence for D_n lattice. 12
1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 13, 13, 7, 1, 1, 25, 55, 25, 9, 1, 1, 41, 169, 147, 41, 11, 1, 1, 61, 411, 625, 309, 61, 13, 1, 1, 85, 853, 2051, 1681, 561, 85, 15, 1, 1, 113, 1583, 5577, 6981, 3721, 923, 113, 17, 1, 1, 145, 2705, 13203, 23673, 18733, 7225, 1415, 145, 19, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Rows 0 and 2 are included by extension since they fit the formula. Row 1 equals the odd numbers in order that triangle A108556 maintains that A108556(n,n-1) = (n/2)*A108556(n,n) for all n>=1, where row n of triangle A108556 equals the inverse binomial transform of row n of this square array.

LINKS

Table of n, a(n) for n=0..65.

FORMULA

T(n, k) = Sum_{j=0..n} C(n+k-j, k-j)*[C(2*n, 2*j) - 2*j*(n-j)*C(n, j)/(n-1)] for n>1, with T(0, k)=1, T(1, k)=2*k+1.

EXAMPLE

Square array begins:

1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...

1,3,5,7,9,11,13,15,17,19,21,23,25,27,...

1,5,13,25,41,61,85,113,145,181,221,265,...

1,13,55,147,309,561,923,1415,2057,2869,...

1,25,169,625,1681,3721,7225,12769,21025,...

1,41,411,2051,6981,18733,42783,86983,...

1,61,853,5577,23673,76389,204205,476113,...

1,85,1583,13203,68853,264825,824083,...

Inverse binomial transform of rows gives

rows of triangle A108556:

1,

1,2,

1,4,4,

1,12,30,20,

1,24,120,192,96,

1,40,330,940,1080,432, ...

Product of the g.f. of row n and (1-x)^(n+1)

generates the symmetric triangle A108558:

1;

1,1;

1,2,1;

1,9,9,1;

1,20,54,20,1;

1,35,180,180,35,1; ...

The row sums of triangle A108558 equals the

main diagonal of triangle A108556.

PROG

(PARI) T(n, k)=if(n<0 || k<0, 0, if(n==0 || k==0, 1, if(n==1, 2*k+1, sum(j=0, k, binomial(n+k-j, k-j)*(binomial(2*n, 2*j)-2*n*binomial(n-2, j-1))))))

CROSSREFS

Cf. A108554 (diagonal), A108555 (antidiagonal sums), A108556, A108558, A001844 (row 2), A005902 (row 3), A007204 (row 4), A008356 (row 5), A008358 (row 6), A008360 (row 7), A008362 (row 8), A008377 (row 9), A008379 (row 10).

Sequence in context: A171229 A125690 A176481 * A176700 A300539 A300966

Adjacent sequences:  A108550 A108551 A108552 * A108554 A108555 A108556

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Jun 10 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 | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 26 17:16 EDT 2020. Contains 334630 sequences. (Running on oeis4.)