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A124725 Triangle read by rows: T(n,k) = binomial(n,k) + binomial(n,k+2) (0 <= k <= n). 2
1, 1, 1, 2, 2, 1, 4, 4, 3, 1, 7, 8, 7, 4, 1, 11, 15, 15, 11, 5, 1, 16, 26, 30, 26, 16, 6, 1, 22, 42, 56, 56, 42, 22, 7, 1, 29, 64, 98, 112, 98, 64, 29, 8, 1, 37, 93, 162, 210, 210, 162, 93, 37, 9, 1, 46, 130, 255, 372, 420, 372, 255, 130, 46, 10, 1, 56, 176, 385, 627, 792, 792, 627 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Binomial transform of the infinite tridiagonal matrix with main diagonal, (1,1,1,...), subdiagonal, (0,0,0,...) and subsubdiagonal, (1,1,1,...). Sum of entries in row n = 2^(n+1) - n - 1 = A000325(n+1).

Riordan array ((1-2x+2x^2)/(1-x)^3, x/(1-x)). - Paul Barry, Apr 08 2011

LINKS

Table of n, a(n) for n=0..72.

FORMULA

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - 3*T(n-2,k) - 2*T(n-2,k-1) + T(n-3,k) + T(n-3,k-1), T(0,0) = T(1,0) = T(1,1) = T(2,2) = 1, T(2,0) = T(2,1) = 2, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Feb 12 2014

EXAMPLE

Row 3 = (4, 4, 3, 1), then row 4 = (7, 8, 7, 4, 1).

First few rows of the triangle are

   1;

   1,  1;

   2,  2,  1;

   4,  4,  3,  1;

   7,  8,  7,  4,  1;

  11, 15, 15, 11,  5,  1;

  16, 26, 30, 26, 16,  6,  1;

  ...

From Paul Barry, Apr 08 2011: (Start)

Production matrix begins

   1, 1;

   1, 1, 1;

   0, 0, 1, 1;

  -1, 0, 0, 1, 1;

   0, 0, 0, 0, 1, 1;

   1, 0, 0, 0, 0, 1, 1;

   0, 0, 0, 0, 0, 0, 1, 1;

  -1, 0, 0, 0, 0, 0, 0, 1, 1;

   0, 0, 0, 0, 0, 0, 0, 0, 1, 1;

   1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1;

(End)

MAPLE

T:=(n, k)->binomial(n, k)+binomial(n, k+2): for n from 0 to 12 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form

MATHEMATICA

Flatten[Table[Binomial[n, k]+Binomial[n, k+2], {n, 0, 20}, {k, 0, n}]] (* Harvey P. Dale, Jun 12 2015 *)

CROSSREFS

Cf. A098574, A000125, A055795, A027660, A055796, A002663.

Cf. A000325.

Sequence in context: A128111 A107356 A329854 * A106522 A128175 A104040

Adjacent sequences:  A124722 A124723 A124724 * A124726 A124727 A124728

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson and Roger L. Bagula, Nov 05 2006

EXTENSIONS

Edited by N. J. A. Sloane, Nov 29 2006

STATUS

approved

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Last modified July 11 17:52 EDT 2020. Contains 335626 sequences. (Running on oeis4.)