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A075513
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Coefficients of Sidi polynomials.
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49
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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; internal format)
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OFFSET
| 1,3
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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(a(n,m)x^m,m=0..n-1), 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(a(n,m)y^m,m=0..n-1), n>=1, are also obtained from diff(((exp(x)-1)^m)/m,x$m)/exp(x) after replacement of exp(x) by y. Here diff(f(x),x$m), m>=1, denotes m-fold differentiation of f(x) with respect to x.
b(k,m,n) := sum((a(m,p)*((p+1)*k)^n)/(m-1)!,p=0..m-1), n>=0, has g.f. 1/product(1-k*p*x,p=1..m) 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.
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REFERENCES
| 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.
A. Sidi, Practical Extrapolation Methods: Theory and Applications, Cambridge University Press, Cambridge, 2003.
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LINKS
| D. S. Lubinsky and H. Stahl, Some Explicit Biorthogonal Polynomials, Paper 181.
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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 (vladeta(AT)eunet.rs), Feb 13 2008
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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.
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CROSSREFS
| Cf. A075510-2, A074932, A075515-16, A075906-25, A076002-13.
Sequence in context: A188922 A036296 A078105 * A011019 A193728 A192424
Adjacent sequences: A075510 A075511 A075512 * A075514 A075515 A075516
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KEYWORD
| sign,tabl,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 02, 2002
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