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A119326 Number triangle T(n,k) = Sum_{j=0..n-k} C(k,2j)*C(n-k,2j). 10
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 7, 10, 7, 1, 1, 1, 1, 11, 19, 19, 11, 1, 1, 1, 1, 16, 31, 38, 31, 16, 1, 1, 1, 1, 22, 46, 66, 66, 46, 22, 1, 1, 1, 1, 29, 64, 106, 126, 106, 64, 29, 1, 1, 1, 1, 37, 85, 162, 226, 226, 162, 85, 37, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,13

COMMENTS

Third column is essentially A000124. Fourth column is essentially A005448. Fifth column is A119327. Product of Pascal's triangle A007318 and A119328. Row sums are A038504. T(n,k) = T(n,n-k).

REFERENCES

Lukas Spiegelhofer and Jeffrey Shallit, Continuants, Run Lengths, and Barry's Modified Pascal Triangle, Volume 26(1) 2019, of The Electronic Journal of Combinatorics, #P1.31.

LINKS

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

Jeffrey Shallit, Lukas Spiegelhofer, Continuants, run lengths, and Barry's modified Pascal triangle, arXiv:1710.06203 [math.CO], 2017.

FORMULA

Column k has g.f.: (x^k/(1-x))* Sum{j=0..k} C(k,2j)*(x/(1-x))^(2j).

T(2n,n) = A119358(n). - Alois P. Heinz, Aug 31 2018

EXAMPLE

Triangle begins:

1;

1, 1;

1, 1,  1;

1, 1,  1,  1;

1, 1,  2,  1,  1;

1, 1,  4,  4,  1,  1;

1, 1,  7, 10,  7,  1, 1;

1, 1, 11, 19, 19, 11, 1, 1;

CROSSREFS

Cf. A119358.

Sequence in context: A026584 A247342 A174547 * A219866 A212363 A212382

Adjacent sequences:  A119323 A119324 A119325 * A119327 A119328 A119329

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, May 14 2006

STATUS

approved

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Last modified December 10 12:30 EST 2019. Contains 329895 sequences. (Running on oeis4.)