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A095801 Square of Narayana triangle A001263: View A001263 as a lower triangular matrix. Then the square of that matrix is also lower triangular. Sequence gives this lower triangle, read by rows. 1
1, 2, 1, 5, 6, 1, 14, 30, 12, 1, 42, 140, 100, 20, 1, 132, 630, 700, 250, 30, 1, 429, 2772, 4410, 2450, 525, 42, 1, 1430, 12012, 25872, 20580, 6860, 980, 56, 1, 4862, 51480, 144144, 155232, 74088, 16464, 1680, 72, 1, 16796, 218790, 772200, 1081080, 698544 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The first three columns are A000108 (the Catalan numbers), A002457 and A085374.
LINKS
FORMULA
T(n, k) = Sum_{i = k..n} A001263(n, i)*A001263(i, k).
T(n, n-1) = n*(n-1).
EXAMPLE
The first 3 rows are 1; 2, 1; 5, 6, 1; since the first 3 rows of the Narayana triangle in matrix format are M = [1 0 0 / 1 1 0 / 1 3 1]. Then M^2 = [1 0 0 / 2 1 0 / 5 6 1].
Triangle starts:
1;
2, 1;
5, 6, 1;
14, 30, 12, 1;
42, 140, 100, 20, 1;
...
MATHEMATICA
t[n_, k_] = Sum[1/(i*k)*(Binomial[i-1, k-1]*Binomial[i, k-1]* Binomial[n-1, i-1]*Binomial[n, i-1]), {i, k, n}];
Flatten[Table[t[n, k], {n, 1, 10}, {k, 1, n}]][[1;; 50]] (* Jean-François Alcover, Jul 21 2011 *)
CROSSREFS
Sequence in context: A107783 A047887 A120986 * A128567 A345394 A217204
KEYWORD
nonn,easy,nice,tabl
AUTHOR
Gary W. Adamson, Jun 07 2004
EXTENSIONS
Edited and extended by David Wasserman, Sep 24 2004
STATUS
approved

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Last modified May 22 15:18 EDT 2024. Contains 372758 sequences. (Running on oeis4.)