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 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 . 0
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 REFERENCES Yang, Sheng-Liang, Yan-Ni Dong, and Tian-Xiao He. "Some matrix identities on colored Motzkin paths." Discrete Mathematics 340.12 (2017): 3081-3091. LINKS Paul Barry, Laurent Biorthogonal Polynomials and Riordan Arrays, arXiv preprint arXiv:1311.2292, 2013 P. Peart and L. Woodson, Triple factorization of some Riordan matrices, The Fib. Quart., Vol. 31 No. 2, May 1993 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 A121576 Adjacent sequences:  A133364 A133365 A133366 * A133368 A133369 A133370 KEYWORD nonn,tabl,easy,changed AUTHOR Philippe Deléham, Oct 27 2007 STATUS approved

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Last modified February 25 00:39 EST 2018. Contains 299630 sequences. (Running on oeis4.)