OFFSET
0,2
COMMENTS
Riordan array ((1-x-sqrt(1-6x+x^2))/(2x), (1-3x-sqrt(1-6x+x^2))/(4x)).
Inverse of Riordan array (1/(1+2x),x/(1+3x+2x^2)) (a signed version of A124237). Paul Barry, Apr 28 2009:
Peart and Woodson give a factorization of this array in the Riordan group as (1/(1 - 3*x), x/(1 - 3*x)) * (C(2*x^2), x*C(2*x^2)) * (1/(1 + x), x), where C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + ... is the o.g.f. of the Catalan numbers A000108. - Peter Bala, Aug 07 2014
LINKS
Paul Barry, Laurent Biorthogonal Polynomials and Riordan Arrays, arXiv preprint arXiv:1311.2292 [math.CA], 2013.
P. Peart and L. Woodson, Triple factorization of some Riordan matrices, The Fib. Quart., Vol. 31 No. 2, May 1993.
Sheng-Liang Yang, Yan-Ni Dong, and Tian-Xiao He, Some matrix identities on colored Motzkin paths, Discrete Mathematics 340.12 (2017): 3081-3091. See p. 3087.
FORMULA
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)+3*T(n-1,k)+2*T(n-1,k+1) for k>=1 .
G.f.: 1/(1-xy-2x-x^2(2+y)/(1-3x-2x^2/(1-3x-2x^2/(1-3x-2x^2/(1- ... (continued fraction). - Paul Barry, Apr 28 2009
Sum_{k, k>=0} T(m,k)*T(n,k)*2^k = T(m+n,0) = A006318(m+n). - Philippe Deléham, Jan 24 2010
T(n,k) = S(n,n-k) - 2*S(n, n-k-2), where S(n,k) = Sum_{j = 0..k} binomial(n-1,k-j)*binomial(n,j)*2^j. - Peter Bala, Feb 20 2018
EXAMPLE
From Paul Barry, Apr 28 2009: (Start)
Triangle begins
1,
2, 1,
6, 5, 1,
22, 23, 8, 1,
90, 107, 49, 11, 1,
394, 509, 276, 84, 14, 1,
1806, 2473, 1505, 556, 128, 17, 1
Production matrix begins
2, 1,
2, 3, 1,
0, 2, 3, 1,
0, 0, 2, 3, 1,
0, 0, 0, 2, 3, 1,
0, 0, 0, 0, 2, 3, 1,
0, 0, 0, 0, 0, 2, 3, 1; (End)
MAPLE
S := proc (n, k)
add(binomial(n-1, k-j)*binomial(n, j)*2^j, j = 0..k);
end proc:
for n from 0 to 10 do
seq(S(n, n-k)-2*S(n, n-k-2), k = 0..n)
end do; # Peter Bala, Feb 20 2018
MATHEMATICA
T[n_, 0] := Hypergeometric2F1[-n, n + 1, 2, -1]; T[n_, k_] := Binomial[-1 + n, -k + n] Hypergeometric2F1[k - n, -n, k, 2] - 2 Binomial[-1 + n, -2 - k + n] Hypergeometric2F1[2 + k - n, -n, 2 + k, 2]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Peter Luschny, Feb 20 2018 *)
CROSSREFS
KEYWORD
AUTHOR
Philippe Deléham, Oct 27 2007
STATUS
approved