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A119301
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Number triangle binomial(3n-k,n-k).
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6
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1, 3, 1, 15, 5, 1, 84, 28, 7, 1, 495, 165, 45, 9, 1, 3003, 1001, 286, 66, 11, 1, 18564, 6188, 1820, 455, 91, 13, 1, 116280, 38760, 11628, 3060, 680, 120, 15, 1, 735471, 245157, 74613, 20349, 4845, 969, 153, 17, 1, 4686825, 1562275, 480700, 134596, 33649, 7315
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OFFSET
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0,2
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COMMENTS
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g(x)=2*sin(arcsin(3*sqrt(3*x)/2)/3)/sqrt(3*x);g(x)=1/(1-xg(x)^2). First column is A005809. Second column is A025174. Row sums are A045721. Inverse is Riordan array (1-3x,x(1-x)^2), A119302.
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LINKS
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FORMULA
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Riordan array (1/(1-3xg(x)^2),xg(x)^2) where g(x)=1+xg(x)^3.
'Horizontal' recurrence equation: T(n,0) = binomial(3*n,n) and for k >= 1, T(n,k) = sum {i = 1..n+1-k} i*T(n-1,k-2+i). - Peter Bala, Dec 28 2014
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EXAMPLE
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Triangle begins
1,
3, 1,
15, 5, 1,
84, 28, 7, 1,
495, 165, 45, 9, 1,
3003, 1001, 286, 66, 11, 1,
18564, 6188, 1820, 455, 91, 13, 1,
116280, 38760, 11628, 3060, 680, 120, 15, 1
...
Horizontal recurrence: T(4,1) = 1*84 + 2*28 + 3*7 + 4*1 = 165. - Peter Bala, Dec 29 2014
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MAPLE
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T := proc(n, k) option remember;
`if`(n = 0, 1, add(i*T(n-1, k-2+i), i=1..n+1-k)) end:
for n from 0 to 9 do print(seq(T(n, k), k=0..n)) od; # Peter Luschny, Dec 30 2014
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MATHEMATICA
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Flatten[Table[Binomial[3n-k, n-k], {n, 0, 10}, {k, 0, n}]] (* Harvey P. Dale, Jul 28 2012 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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