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A091811
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Array read by rows: T(n,k) = binomial(n+k-2,k-1)*binomial(2*n-1,n-k).
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1
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1, 3, 2, 10, 15, 6, 35, 84, 70, 20, 126, 420, 540, 315, 70, 462, 1980, 3465, 3080, 1386, 252, 1716, 9009, 20020, 24024, 16380, 6006, 924, 6435, 40040, 108108, 163800, 150150, 83160, 25740, 3432, 24310, 175032, 556920, 1021020, 1178100, 875160
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OFFSET
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1,2
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COMMENTS
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Alternating sum of elements of n-th row = 1.
If a certain event has a probability p of occurring in any given trial, the probability of its occurring at least n times in 2n-1 trials is Sum_{k=1..n} T(n,k)*(-1)^(k-1)*p^(n+k-1). For example, the probability of its occurring at least 4 out of 7 times is 35p^4 - 84p^5 + 70p^6 - 20p^7. - Matthew Vandermast, Jun 05 2004
With the row polynomial defined as R(n,x) = Sum_{k = 1..n} T(n,k)*x^k, the row polynomial is related to the regularized incomplete Beta function I_x(a,b), through the relation R(n,x) = -(-x)^{-n+1}*I_{-x}(n,n). - Leo C. Stein, Jun 06 2019
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LINKS
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FORMULA
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O.g.f.: x*t*(1+2*x-sqrt(1-4*t*(x+1)))/(2*(x+t)*sqrt(1-4*t*(x+1))) = x*t + (3*x+2*x^2)*t^2 + (10*x+15*x^2+6*x^3)*t^3 + ....
Sum_{k = 1..n} (-1)^(k-1)*T(n,k)*2^(n-k) = 4^(n-1).
Row polynomial R(n+1,x) = ((2*n+1)!/n!^2)*x*Integral_{y = 0..1} (y*(1+x*y))^n dy. Row sums A178792. (End)
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EXAMPLE
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Triangle starts:
1,
3, 2,
10, 15, 6,
35, 84, 70, 20,
126, 420, 540, 315, 70,
...
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MATHEMATICA
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t[n_, k_] := Binomial[n+k-2, k-1]*Binomial[2n-1, n-k]; Table[t[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 06 2012 *)
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PROG
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(PARI) T(x, y)=binomial(x+y-2, y-1)*binomial(2*x-1, x-y)
(Magma) [[Binomial(n+k-2, k-1)*Binomial(2*n-1, n-k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Jun 15 2015
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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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