1,2

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

From Peter Bala, Apr 10 2012: (Start)

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)

Triangle starts:

1,

3, 2,

10, 15, 6,

35, 84, 70, 20,

126, 420, 540, 315, 70,

...

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 *)

(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

Cf. A001700 (first column), A002740 (second column), A000984 (main diagonal), A033876 (second diagonal), A178792 (row sums).

Sequence in context: A300374 A256063 A006743 * A327809 A075856 A025520

Adjacent sequences: A091808 A091809 A091810 * A091812 A091813 A091814

nonn,tabl,nice

Benoit Cloitre, Mar 18 2004

approved