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A134636 Triangle formed by Pascal's rule given borders = 2n+1. 7
1, 3, 3, 5, 6, 5, 7, 11, 11, 7, 9, 18, 22, 18, 9, 11, 27, 40, 40, 27, 11, 13, 38, 67, 80, 67, 38, 13, 15, 51, 105, 147, 147, 105, 51, 15, 17, 66, 156, 252, 294, 252, 156, 66, 17, 19, 83, 222, 408, 546, 546, 408, 222, 83, 19, 21, 102, 305, 630, 954, 1092, 954, 630, 305, 102, 21 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Row sums = A048487: (1, 6, 16, 36, 76, 156,...).

T(n,k) = A051601(n,k) + A051597(n,k); T(n,k) mod 2 = A047999(n,k). - Reinhard Zumkeller, Nov 23 2012

LINKS

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

Index entries for triangles and arrays related to Pascal's triangle

FORMULA

Triangle, given borders = (1, 3, 5, 7, 9,...); apply Pascal's rule T(n,k) = T(n-1,k) P T(n-1,k-1).

Closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - Boris Putievskiy, Aug 19 2013

EXAMPLE

First few rows of the triangle are:

1;

3, 3;

5, 6, 5;

7, 11, 11, 7;

9, 18, 22, 18, 9;

11, 27, 40, 40, 27, 11;

13, 38, 67, 80, 67, 38, 13;

...

MAPLE

T:= proc(n, k) option remember;

      `if`(k<0 or k>n, 0,

      `if`(k=0 or k=n, 2*n+1,

         T(n-1, k-1) + T(n-1, k) ))

    end:

seq(seq(T(n, k), k=0..n), n=0..14);  # Alois P. Heinz, May 26 2013

MATHEMATICA

NestList[Append[

    Prepend[Map[Apply[Plus, #] &, Partition[#, 2, 1]], #[[1]] +

2], #[[1]] + 2] &, {1}, 10] // Grid  (* Geoffrey Critzer, May 26 2013 *)

PROG

(Haskell)

a134636 n k = a134636_tabl !! n !! k

a134636_row n = a134636_tabl !! n

a134636_tabl = iterate (\row -> zipWith (+) ([2] ++ row) (row ++ [2])) [1]

-- Reinhard Zumkeller, Nov 23 2012

CROSSREFS

Cf. A048487.

Sequence in context: A317745 A204250 A131948 * A088564 A161560 A078796

Adjacent sequences:  A134633 A134634 A134635 * A134637 A134638 A134639

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Nov 04 2007

EXTENSIONS

Offset changed by Reinhard Zumkeller, Nov 23 2012

STATUS

approved

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Last modified September 20 03:53 EDT 2019. Contains 327210 sequences. (Running on oeis4.)