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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

Table of n, a(n) for n=1..39.

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-Fran├žois 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,changed

AUTHOR

Wolfdieter Lang, Oct 02 2002

STATUS

approved

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Last modified May 26 18:13 EDT 2018. Contains 304630 sequences. (Running on oeis4.)