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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
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OFFSET
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0,2
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COMMENTS
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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, Mar 14 2007
Central coefficients T(2n,n) are binomial(4n,n) (A005810).
The A- and Z-sequence for this Riordan triangle is [1,2,1] and [2,2], respectively. For the notion of Z- and A-sequences for Riordan arrays see the W. Lang link under A006232 with details and references. See also the Philippe DELEHAM comment above. - Wolfdieter Lang, Nov 22 2012
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REFERENCES
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Asamoah Nkwanta and Earl R. Barnes, Two Catalan-type Riordan Arrays and their Connections to the Chebyshev Polynomials of the First Kind, Journal of Integer Sequences, Article 12.3.3, 2012. - From N. J. A. Sloane, Sep 16 2012
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LINKS
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Table of n, a(n) for n=0..56.
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FORMULA
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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, 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, Jul 14 2005
T(n,k)=sum{j=0..n, C(n,j)C(n,j-k)}. - Paul Barry, Mar 07 2006
T(n,k)= Sum_{h>=k}A039599(n,h) . Sum_{0<=k<=n}T(n,k)= A032443(n) . - Philippe Deléham, May 01 2006
Sum_{k = 0...n}T(n,k)^2 = A036910(n) . - Philippe Deléham, May 07 2006
Sum_{k, 0<=k<=n}T(n,k)*(-1)^k=A088218(n) . - Philippe DELEHAM, Mar 14 2007
From Wolfdieter Lang, Nov 22 2012 (Start)
The o.g.f. for the row polynomials P(n,x) := sum(T(n,k)*x^k, k=0..n) is G(z,x) = (-x + (1+x)*z + x*z*c(z))/(sqrt(1-4*z)*((1+x)^2*z -x)) with c the o.g.f. of A000108 (Catalan). This follows from the Riordan property.
The o.g.f. for column No. k is (c(x)-1)^k/sqrt(1-4*x) (from the Riordan property). (End)
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EXAMPLE
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The triangle T(n,k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 10
0: 1
1: 2 1
2: 6 4 1
3: 20 15 6 1
4: 70 56 28 8 1
5: 252 210 120 45 10 1
6: 924 792 495 220 66 12 1
7: 3432 3003 2002 1001 364 91 14 1
8: 12870 11440 8008 4368 1820 560 120 16 1
9: 48620 43758 31824 18564 8568 3060 816 153 18 1
10: 184756 167960 125970 77520 38760 15504 4845 1140 190 20 1
... Reformatted ad extended by Wolfdieter Lang. Nov 22 2012
Contribution from Paul Barry, 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)
From Wolfdieter Lang, Nov 22 2012 (Start)
Recurrence from the Riordan A-sequence [1,2,1]: T(4,1) = 56 =
1*T(3,0) + 2*T(3,1) + 1*T(3,2) = 1*20 + 2*15 + 1*6.
Recurrence from the Riordan Z-sequence [2,2]: T(7,0) = 3432 = 2*T(6,0) + 2*T(6,1) = 2*924 + 2*792. See the Philippe DELEHAM comment above. (End)
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CROSSREFS
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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
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easy,nonn,tabl
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AUTHOR
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Paul Barry, May 07 2004
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EXTENSIONS
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Entry revised by N. J. A. Sloane, Mar 23 2007
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STATUS
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approved
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