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 A062991 Coefficient triangle for certain polynomials N(2; n,x) (rising powers of x). 13
 1, 2, -1, 5, -6, 2, 14, -28, 20, -5, 42, -120, 135, -70, 14, 132, -495, 770, -616, 252, -42, 429, -2002, 4004, -4368, 2730, -924, 132, 1430, -8008, 19656, -27300, 23100, -11880, 3432, -429, 4862, -31824, 92820, -157080, 168300, -116688, 51051, -12870, 1430 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The g.f. for the sequence of column m of triangle A009766(n,m) (or Catalan A033184(n,n-m) diagonals) is N(2; m-1,x)*(x^m)/(1-x)^(m+1), m >= 1, with N(2; n,x) := sum(a(n,k)*x^k,k=0..n). For k=0..1 the column sequences give A000108(n+1) (Catalan), -A002694. The row sums give A000012 (powers of 1) and (unsigned) A062992. Another version of [1, 1, 1, 1, 1, 1, 1, 1, ...] DELTA [0, -1, -1, -1, -1, -1, -1, -1, ...] = 1; 1, 0; 2, -1, 0; 5, -6, 2, 0; 14, -28, 20, -5, 0; 42, -120, 135, -70, 14, 0; ... where DELTA is Deléham's operator defined in A084938. The positive triangle has T(n,k)=binomial(2n+2,n-k)*binomial(n+k,k)/(n+1). - Paul Barry, May 11 2005 LINKS C. A. Francisco, J. Mermin, J. Schweig, Catalan numbers, binary trees, and pointed pseudotriangulations, 2013. A. Lakshminarayan, Z. Puchala, K. Zyczkowski, Diagonal unitary entangling gates and contradiagonal quantum states, arXiv preprint arXiv:1407.1169, 2014 FORMULA a(n, k) := [x^k]N(2; n, x) with N(2; n, x)=(N(2; n-1, x)-A000108(n)*(1-x)^(n+1))/x, N(2; 0, x) := 1. a(n, k)= a(n-1, k+1)+((-1)^k)*binomial(n+1, k+1)*binomial(2*n+1, n)/(2*n+1) if k=0, .., (n-2); a( n, k)= ((-1)^k)*binomial(n+1, k+1)*binomial(2*n+1, n)/(2*n+1) if k=(n-1) or n; else 0. O.g.f. (with offset 1) is the series reversion w.r.t. x of x*(1+x*t)/(1+x)^2. If R(n,t) denotes the n-th row polynomial of this triangle then R(n,1-t) is the n-th row polynomial of A009766. Cf. A089434. EXAMPLE {1}; {2,-1}; {5,-6,2}; {14,-28,20,-5}; ...; N(2; 2,x)=5-6*x+2*x^2. MATHEMATICA T[n_, k_] := 2 (-1)^k Binomial[2n+1, n] (n-k+1) Binomial[n+1, k]/((k+n+1)(k+n+2)); Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 19 2018 *) CROSSREFS Cf. A009766, A089434. For an unsigned version see Borel's triangle, A234950. Sequence in context: A231732 A185384 A274728 * A234950 A275228 A118984 Adjacent sequences:  A062988 A062989 A062990 * A062992 A062993 A062994 KEYWORD sign,easy,tabl AUTHOR Wolfdieter Lang, Jul 12 2001 STATUS approved

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Last modified October 19 15:19 EDT 2019. Contains 328223 sequences. (Running on oeis4.)