A133367 Triangle T(n,k) read by rows given by [2,1,2,1,2,1,2,1,2,1,2,1,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938 .

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, 8558, 12235, 8100, 3429, 974, 181, 20, 1, 41586, 61463, 43393, 20355, 6713, 1557, 243, 23, 1

Offset: 0

Author

Philippe Deléham, Oct 27 2007

Keywords

• nonn
• tabl
• easy

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

Examples

			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)
		

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.

Crossrefs

Cf. A006318, A000012, A016789.

Programs

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 *)

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