A133336 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,1,1,1,1,1,1,...] DELTA [0,1,0,1,0,1,0,1,0,...] where DELTA is the operator defined in A084938.

1, 1, 0, 2, 1, 0, 5, 5, 1, 0, 14, 21, 9, 1, 0, 42, 84, 56, 14, 1, 0, 132, 330, 300, 120, 20, 1, 0, 429, 1287, 1485, 825, 225, 27, 1, 0, 1430, 5005, 7007, 5005, 1925, 385, 35, 1, 0, 4862, 19448, 32032, 28028, 14014, 4004, 616, 44, 1, 0, 16796, 75582, 143208, 148512, 91728, 34398, 7644, 936, 54, 1, 0

Offset

0,4

Comments

Mirror image of triangle A086810; another version of A126216.

Equals A131198*A007318 as infinite lower triangular matrices. - Philippe Deléham, Oct 23 2007

Diagonal sums: A119370. - Philippe Deléham, Nov 09 2009

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

W. Y. C. Chen, T. Mansour and S. H. F. Yan, Matchings avoiding partial patterns, The Electronic Journal of Combinatorics 13, 2006, #R112, Theorem 3.3.

Formula

Sum_{k=0..n} T(n,k)*x^k = A000108(n), A001003(n), A007564(n), A059231(n), A078009(n), A078018(n), A081178(n), A082147(n), A082181(n), A082148(n), A082173(n) for x = 0,1,2,3,4,5,6,7,8,9,10 respectively.

Sum_{k=0..n} T(n,k)*x^(n-k) = A000007(n), A001003(n), A107841(n), A131763(n), A131765(n), A131846(n), A131926(n), A131869(n), A131927(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively. - Philippe Deléham, Nov 05 2007

Sum_{k=0..n} T(n,k)*(-2)^k*5^(n-k) = A152601(n). - Philippe Deléham, Dec 10 2008

T(n,k) = binomial(n-1,k)*binomial(2n-k,n)/(n+1), k <= n. - Philippe Deléham, Nov 02 2009

Example

Triangle begins:
    1;
    1,    0;
    2,    1,    0;
    5,    5,    1,   0;
   14,   21,    9,   1,   0;
   42,   84,   56,  14,   1,  0;
  132,  330,  300, 120,  20,  1, 0;
  429, 1287, 1485, 825, 225, 27, 1, 0;

Mathematica

Table[Binomial[n-1, k]*Binomial[2*n-k, n]/(n+1), {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, feb 05 2018 *)

Prog

(PARI) for(n=0, 10, for(k=0, n, print1(binomial(n-1, k)*binomial(2*n-k, n)/(n+1), ", "))) \\ G. C. Greubel, Feb 05 2018
(Magma) [[Binomial(n-1, k)*Binomial(2*n-k, n)/(n+1): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Feb 05 2018

Crossrefs

Cf. A000108, A002054, A002055, A002056, A007160, A033280, A033281, A033282.

Sequence in context: A182376 A030206 A212768 * A269951 A176056 A298213

Adjacent sequences: A133333 A133334 A133335 * A133337 A133338 A133339

Keyword

nonn,tabl

Author

Philippe Deléham, Oct 19 2007

Status

approved