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 A075513 Coefficients of Sidi polynomials. 52
 1, -1, 2, 1, -8, 9, -1, 24, -81, 64, 1, -64, 486, -1024, 625, -1, 160, -2430, 10240, -15625, 7776, 1, -384, 10935, -81920, 234375, -279936, 117649, -1, 896, -45927, 573440, -2734375, 5878656, -5764801, 2097152, 1, -2048, 183708 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Coefficients of the Sidi polynomials D(a,b)_n(x) when a = b = 0. See [Sidi 1980]. The row polynomials p(n,x) := Sum_{m=0..n-1} a(n,m)x^m, n >= 1, are obtained from ((Eu(x)^n)*(x-1)^n)/(n*x), where Eu(x) := xd/dx is the Euler-derivative with respect to x. The row polynomials p(n,y) := Sum_{m=0..n-1} a(n,m)y^m, n >= 1, are also obtained from ((d^m/dx^m)((exp(x)-1)^m)/m)/exp(x) after replacement of exp(x) by y. Here (d^m/dx^m)f(x), m >= 1, denotes m-fold differentiation of f(x) with respect to x. b(k,m,n) := (Sum_{p=0..m-1} (a(m,p)*((p+1)*k)^n))/(m-1)!, n >= 0, has g.f. 1/Product_{p=1..m} (1 - k*p*x) for k=1,2,... and m=1,2,... The (signed) row sums give A000142(n-1), n >= 1, (factorials) and (unsigned) A074932(n). The (unsigned) columns give A000012 (powers of 1), 2*A001787(n+1), (3^2)*A027472(n), (4^3)*A038846(n-1), (5^4)A036071(n-5), (6^5)*A036084(n-6), (7^6)* A036226(n-7), (8^7)*A053107(n-8) for m=0..7. Right edge of triangle is A000169. - Michel Marcus, May 17 2013 REFERENCES A. Sidi, Practical Extrapolation Methods: Theory and Applications, Cambridge University Press, Cambridge, 2003. LINKS D. S. Lubinsky and H. Stahl, Some Explicit Biorthogonal Polynomials, (IN) Approximation Theory XI, (C.K. Chui, M. Neamtu, L. Schumaker, eds.), Nashboro Press, Nashville, 2005, pp. 279-285. A. Sidi, Numerical Quadrature and Non-Linear Sequence Transformations: Unified Rules for Efficient Computation of Integrals with Algebraic and Logarithmic Endpoint Singularities, Math. Comp., 35(1980), 851-874. FORMULA a(n, m) = ((-1)^(n-m-1)) binomial(n-1, m)*(m+1)^(n-1), n >= m+1 >= 1, else 0. G.f. for m-th column: ((m+1)^m)(x/(1+(m+1)*x))^(m+1), m >= 0. E.g.f.: -LambertW(-x*y*exp(-x))/(1+LambertW(-x*y*exp(-x)))/x. - Vladeta Jovovic, Feb 13 2008 a(n, k) = T(n, k+1) / n where T(, ) is triangle in A258773. - Michael Somos, May 13 2018 EXAMPLE [ 1]; [-1,   2]; [ 1,  -8,   9]; [-1,  24, -81,  64]; ... p(2,x) = -1+2*x = (1/(2*x))*x*(d/dx)*x*(d/dx)*(x-1)^2. MATHEMATICA p[n_, x_] := p[n, x] = Nest[ x*D[#, x]& , (x-1)^n, n]/(n*x); a[n_, m_] := Coefficient[ p[n, x], x, m]; Table[a[n, m], {n, 1, 9}, {m, 0, n-1}] // Flatten (* Jean-François Alcover, Jul 03 2013 *) PROG (PARI) tabl(nn) = {for (n=1, nn, for (m=0, n-1, print1((-1)^(n-m-1)*binomial(n-1, m)*(m+1)^(n-1), ", "); ); print(); ); } \\ Michel Marcus, May 17 2013 CROSSREFS Cf. A075510, A075511, A075512, A074932, A075515, A075516, A075906..A075925, A076002..A076013. Cf. A258773. Sequence in context: A188922 A036296 A078105 * A284211 A246403 A258502 Adjacent sequences:  A075510 A075511 A075512 * A075514 A075515 A075516 KEYWORD sign,tabl,easy AUTHOR Wolfdieter Lang, Oct 02 2002 STATUS approved

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Last modified January 22 01:28 EST 2022. Contains 350481 sequences. (Running on oeis4.)