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 A185296 Triangle of connection constants between the falling factorials (x)_(n) and (2*x)_(n). 0
 1, 0, 2, 0, 2, 4, 0, 0, 12, 8, 0, 0, 12, 48, 16, 0, 0, 0, 120, 160, 32, 0, 0, 0, 120, 720, 480, 64, 0, 0, 0, 0, 1680, 3360, 1344, 128, 0, 0, 0, 0, 1680, 13440, 13440, 3584, 256, 0, 0, 0, 0, 0, 30240, 80640, 48384, 9216, 512 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The falling factorial polynomials (x)_n := x*(x-1)*...*(x-n+1), n = 0,1,2,..., form a basis for the space of polynomials. Hence the polynomial (2*x)_n may be expressed as a linear combination of x_0, x_1,...,x_n; the coefficients in the expansion form the n-th row of the table. Some examples are given below. This triangle is connected to two families of orthogonal polynomials, the Hermite polynomials H(n,x) A060821, and the Bessel polynomials y(n,x)A001498. The first few Hermite polynomials are ... H(0,x) = 1 ... H(1,x) = 2*x ... H(2,x) = -2+4*x^2 ... H(3,x) = -12*x+8*x^3 ... H(4,x) = 12-48*x^2+16*x^4. The unsigned coefficients of H(n,x) give the nonzero entries of the n-th row of the triangle. The Bessel polynomials y(n,x) begin ... y(0,x) = 1 ... y(1,x) = 1+x ... y(2,x) = 1+3*x+3*x^2 ... y(3,x) = 1+6*x+15*x^2+15*x^3. The entries in the n-th column of this triangle are the coefficients of the scaled Bessel polynomials 2^n*y(n,x). Also the Bell transform of g(n) = 2 if n<2 else 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016 REFERENCES L. Comtet, Advanced Combinatorics, Reidel, 1974, page 158, exercise 7. LINKS FORMULA Defining relation: 2*x*(2*x-1)*...*(2*x-n+1) = sum {k=0..n} T(n, k)*x*(x-1)*...*(x-k+1) Explicit formula: T(n,k) = (n!/k!)*binomial(k,n-k)*2^(2*k-n). [As defined by Comtet (see reference).] Recurrence relation: T(n,k) = (2*k-n+1)*T(n-1,k)+2*T(n-1,k-1). E.g.f.: exp(x*(t^2+2*t)) = 1 + (2*x)*t + (2*x+4*x^2)*t^2/2! + (12*x^2+8*x^3)*t^3/3! + ... O.g.f. for m-th diagonal (starting at main diagonal m = 0): (2*m)!/m!*x^m/(1-2*x)^(2*m+1). The triangle is the matrix product [2^k*s(n,k)]n,k>=0 * ([s(n,k)]n,k>=0)^(-1), where s(n,k) denotes the signed Stirling number of the first kind. Row sums are [1,2,6,20,76,...] = A000898. Column sums are [1,4,28,296,...] = [2^n*A001515(n)] n>=0. EXAMPLE Triangle begins n\k|...0.....1.....2.....3.....4.....5.....6 ============================================ 0..|...1 1..|...0.....2 2..|...0.....2.....4 3..|...0.....0....12.....8 4..|...0.....0....12....48....16 5..|...0.....0.....0...120...160....32 6..|...0.....0.....0...120...720...480....64 .. Row 3: (2*x)_3 = (2*x)*(2*x-1)*(2*x-2) = 8*x*(x-1)*(x-2) + 12*x*(x-1). Row 4: (2*x)_4 = (2*x)*(2*x-1)*(2*x-2)*(2*x-3) = 16*x*(x-1)*(x-2)*(x-3) + 48*x*(x-1)*(x-2)+ 12*x*(x-1). Examples of recurrence relation T(4,4) = 5*T(3,4) + 2*T(3,3) = 5*0 + 2*8 = 16; T(5,4) = 4*T(4,4) + 2*T(4,3) = 4*16 + 2*48 = 160; T(6,4) = 3*T(5,4) + 2*T(5,3) = 3*160 + 2*120 = 720; T(7,4) = 2*T(6,4) + 2*T(6,3) = 2*720 + 2*120 = 1680. MAPLE T := (n, k) -> (n!/k!)*binomial(k, n-k)*2^(2*k-n): seq(seq(T(n, k), k=0..n), n=0..9); PROG (Sage) # The function bell_matrix is defined in A264428. bell_matrix(lambda n: 2 if n<2 else 0, 12) # Peter Luschny, Jan 19 2016 CROSSREFS Cf. A000898 (row sums), A001498, A001515, A059343, A060821. Sequence in context: A151668 A086151 A099040 * A136717 A261685 A136716 Adjacent sequences:  A185293 A185294 A185295 * A185297 A185298 A185299 KEYWORD nonn,easy,tabl AUTHOR Peter Bala, Feb 20 2011 STATUS approved

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Last modified October 17 22:48 EDT 2018. Contains 316297 sequences. (Running on oeis4.)