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A092392
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Triangle read by rows: T(n,k) = C(2*n - k,n), 0 <= k <= n.
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29
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1, 2, 1, 6, 3, 1, 20, 10, 4, 1, 70, 35, 15, 5, 1, 252, 126, 56, 21, 6, 1, 924, 462, 210, 84, 28, 7, 1, 3432, 1716, 792, 330, 120, 36, 8, 1, 12870, 6435, 3003, 1287, 495, 165, 45, 9, 1, 48620, 24310, 11440, 5005, 2002, 715, 220, 55, 10, 1, 184756, 92378, 43758, 19448, 8008, 3003, 1001, 286, 66, 11, 1
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OFFSET
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0,2
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COMMENTS
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Viewed as the square array [binomial (2*n + k, n + k)]n,k>=0 this is the generalized Riordan array ( 1/sqrt(1 - 4*x),c(x) ) in the sense of the Bala link, where c(x) is the o.g.f. for A000108.
The square array factorizes as ( 1/(2 - c(x)),x*c(x) ) * ( 1/(1 - x),1/(1 - x) ), which equals the matrix product of A100100 with the square Pascal matrix [binomial (n + k,k)]n,k>=0. See the example below. (End)
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LINKS
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FORMULA
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As a number triangle, this is T(n, k) = if(k <= n, C(2*n - k, n), 0). Its row sums are C(2*n + 1, n + 1) = A001700. Its diagonal sums are A176287. - Paul Barry, Apr 23 2005
G.f. of column k: 2^k/[sqrt(1 - 4*x)*(1 + sqrt(1 - 4*x))^k].
As a number triangle, this is the Riordan array (1/sqrt(1 - 4*x), x*c(x)), c(x) the g.f. of A000108. - Paul Barry, Jun 24 2005
G.f.: A(x,y)=1/sqrt(1 - 4*x)/(1-y*x*C(x)), where C(x) is g.f. of Catalan numbers. - Vladimir Kruchinin, Mar 19 2016
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EXAMPLE
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Triangle begins
1,
2, 1,
6, 3, 1,
20, 10, 4, 1,
70, 35, 15, 5, 1,
252, 126, 56, 21, 6, 1,
924, 462, 210, 84, 28, 7, 1,
3432, 1716, 792, 330, 120, 36, 8, 1
Production array is
2, 1,
2, 1, 1,
2, 1, 1, 1,
2, 1, 1, 1, 1,
2, 1, 1, 1, 1, 1,
2, 1, 1, 1, 1, 1, 1,
2, 1, 1, 1, 1, 1, 1, 1,
2, 1, 1, 1, 1, 1, 1, 1, 1,
2, 1, 1, 1, 1, 1, 1, 1, 1, 1 (End)
As a square array = A100100 * square Pascal matrix:
/1 1 1 1 ...\ / 1 \/1 1 1 1 ...\
|2 3 4 5 ...| | 1 1 ||1 2 3 4 ...|
|6 10 15 21 ...| = | 3 2 1 ||1 3 6 10 ...|
|20 35 56 84 ...| |10 6 3 1 ||1 4 10 20 ...|
|70 ... | |35 ... ||1 ... |
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MAPLE
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binomial(2*n-k, n-k) ;
end proc:
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MATHEMATICA
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Table[Binomial[2 n - k, n], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Mar 19 2016 *)
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PROG
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(Haskell)
a092392 n k = a092392_tabl !! (n-1) !! (k-1)
a092392_row n = a092392_tabl !! (n-1)
a092392_tabl = map reverse a046899_tabl
(Maxima)
C(x):=(1-sqrt(1-4*x))/2;
A(x, y):=(1/sqrt(1-4*x))/(1-y*C(x));
(PARI) for(n=0, 10, for(k=0, n, print1(binomial(2*n - k, n), ", "))) \\ G. C. Greubel, Nov 22 2017
(Magma) /* As a triangle */ [[Binomial(2*n-k, n): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 22 2017
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CROSSREFS
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Rows 0-14 are A000984, A001700, A001791, A002054, A002694, A003516, A002696, A030053, A004310, A030054, A004311, A030055, A004312, A030056, A004313.
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KEYWORD
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AUTHOR
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EXTENSIONS
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Diagonal sums comment corrected by Paul Barry, Apr 14 2010
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STATUS
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approved
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