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A094527
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Triangle T(n,k), read by rows, defined by T(n,k)= binomial(2*n,n-k).
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7
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1, 2, 1, 6, 4, 1, 20, 15, 6, 1, 70, 56, 28, 8, 1, 252, 210, 120, 45, 10, 1, 924, 792, 495, 220, 66, 12, 1, 3432, 3003, 2002, 1001, 364, 91, 14, 1, 12870, 11440, 8008, 4368, 1820, 560, 120, 16, 1, 48620, 43758, 31824, 18564, 8568, 3060, 816, 153, 18, 1, 184756, 167960
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Right-hand side of even-numbered rows of Pascal's triangle.
Row sums are A032443. Reverse of A062344. Right-hand side of A034870. Binomial transform of trinomial triangle A094531.
Triangle T(n,k),0<=k<=n, read by rows defined by :T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=2*T(n-1,0)+2*T(n-1,1), T(n,k)=T(n-1,k-1)+2*T(n-1,k)+T(n-1,k+1) for k>=1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 14 2007
Central coefficients T(2n,n) are binomial(4n,n) (A005810).
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FORMULA
| Riordan array (1/sqrt(1-4x)), (1-2x-sqrt(1-4x))/(2x)). Column k has e.g.f. exp(2x)Bessel_I(k, 2x). - Paul Barry (pbarry(AT)wit.ie), Jul 14 2005
Product of Product of Riordan arrays (1/(1-x), x/(1-x)) (Pascal's triangle, A007318) and (1/sqrt(1-2x-3x^2), (1-x-sqrt(1-2x-3x^2))/(2x)) (A094531). Inverse is A110162. - Paul Barry (pbarry(AT)wit.ie), Jul 14 2005
T(n,k)=sum{j=0..n, C(n,j)C(n,j-k)}. - Paul Barry (pbarry(AT)wit.ie), Mar 07 2006
T(n,k)= Sum_{h>=k}A039599(n,h) . Sum_{0<=k<=n}T(n,k)= A032443(n) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), May 01 2006
Sum_{k = 0...n}T(n,k)^2 = A036910(n) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), May 07 2006
Sum_{k, 0<=k<=n}T(n,k)*(-1)^k=A088218(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 14 2007
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EXAMPLE
| Triangle begins:
1;
2, 1;
6, 4, 1;
20, 15, 6, 1;
70, 56, 28, 8, 1;
252, 210, 120, 45, 10, 1;
924, 792, 495, 220, 66, 12, 1;
Contribution from Paul Barry (pbarry(AT)wit.ie), Sep 07 2009: (Start)
Production array is
2, 1,
2, 2, 1,
0, 1, 2, 1,
0, 0, 1, 2, 1,
0, 0, 0, 1, 2, 1,
0, 0, 0, 0, 1, 2, 1,
0, 0, 0, 0, 0, 1, 2, 1 (End)
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CROSSREFS
| Cf. A007318, A032443, A039599.
Sequence in context: A118040 A073387 A125693 * A054335 A110681 A117852
Adjacent sequences: A094524 A094525 A094526 * A094528 A094529 A094530
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), May 07 2004
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EXTENSIONS
| Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Mar 23 2007
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