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

0,2

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

Table of n, a(n) for n=0..44.

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

Cf. A006318, A000012, A016789.

Sequence in context: A159965 A116395 A159924 * A179456 A214152 A121575

Adjacent sequences: A133364 A133365 A133366 * A133368 A133369 A133370

Keyword

nonn,tabl,easy

Author

Philippe Deléham, Oct 27 2007

Status

approved