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 A075513 Coefficients of Sidi polynomials. 52

%I

%S 1,-1,2,1,-8,9,-1,24,-81,64,1,-64,486,-1024,625,-1,160,-2430,10240,

%T -15625,7776,1,-384,10935,-81920,234375,-279936,117649,-1,896,-45927,

%U 573440,-2734375,5878656,-5764801,2097152,1,-2048,183708

%N Coefficients of Sidi polynomials.

%C Coefficients of the Sidi polynomials D(a,b)_n(x) when a = b = 0. See [Sidi 1980].

%C 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.

%C 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.

%C 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,...

%C The (signed) row sums give A000142(n-1), n >= 1, (factorials) and (unsigned) A074932(n).

%C 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.

%C Right edge of triangle is A000169. - _Michel Marcus_, May 17 2013

%D A. Sidi, Practical Extrapolation Methods: Theory and Applications, Cambridge University Press, Cambridge, 2003.

%H D. S. Lubinsky and H. Stahl, <a href="http://people.math.gatech.edu/~lubinsky/Research%20papers/GatlinburgProcRohrsVn.pdf">Some Explicit Biorthogonal Polynomials</a>, (IN) Approximation Theory XI, (C.K. Chui, M. Neamtu, L. Schumaker, eds.), Nashboro Press, Nashville, 2005, pp. 279-285.

%H A. Sidi, <a href="https://doi.org/10.1090/S0025-5718-1980-0572861-2">Numerical Quadrature and Non-Linear Sequence Transformations: Unified Rules for Efficient Computation of Integrals with Algebraic and Logarithmic Endpoint Singularities</a>, Math. Comp., 35(1980), 851-874.

%F a(n, m) = ((-1)^(n-m-1)) binomial(n-1, m)*(m+1)^(n-1), n >= m+1 >= 1, else 0.

%F G.f. for m-th column: ((m+1)^m)(x/(1+(m+1)*x))^(m+1), m >= 0.

%F E.g.f.: -LambertW(-x*y*exp(-x))/(1+LambertW(-x*y*exp(-x)))/x. - _Vladeta Jovovic_, Feb 13 2008

%F a(n, k) = T(n, k+1) / n where T(, ) is triangle in A258773. - _Michael Somos_, May 13 2018

%e [ 1];

%e [-1, 2];

%e [ 1, -8, 9];

%e [-1, 24, -81, 64]; ...

%e p(2,x) = -1+2*x = (1/(2*x))*x*(d/dx)*x*(d/dx)*(x-1)^2.

%t 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 *)

%o (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

%Y Cf. A075510, A075511, A075512, A074932, A075515, A075516, A075906..A075925, A076002..A076013.

%Y Cf. A258773.

%K sign,tabl,easy

%O 1,3

%A _Wolfdieter Lang_, Oct 02 2002

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Last modified March 30 19:43 EDT 2020. Contains 333127 sequences. (Running on oeis4.)