A075513 Triangle read by rows. T(n, m) are the coefficients of Sidi polynomials.

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, -3670016, 27343750, -94058496, 161414428, -134217728, 43046721

Offset

1,3

Comments

Coefficients of the Sidi polynomials (-1)^(n-1)*D_{n-1,1,n-1}(x), for n >=1, where D_{k,n,m}(z) is given in Theorem 4.2., p. 862, of 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..45.

Wolfdieter Lang, On a Certain Family of Sidi Polynomials, May 2023.

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.

Avram Sidi, Numerical Quadrature and Nonlinear Sequence Transformations; Unified Rules for Efficient Computation of Integrals with Algebraic and Logarithmic Endpoint Singularities, Math. Comp., 35 (1980), 851-874.

Formula

T(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*y). - Vladeta Jovovic, Feb 13 2008 [corrected for offset 0 <= m <= n. For offset n >= 1 take the integral over x. - Wolfdieter Lang, Oct 12 2022]

T(n, k) = S(n, k+1) / n where S(, ) is triangle in A258773. - Michael Somos, May 13 2018

E.g.f. of column k, with offset n >= 0: exp(-(k + 1)*x)*((k + 1)*x)^k/k!. - Wolfdieter Lang, Oct 20 2022

E.g.f: 1/(exp(LambertW(-exp(-x)*x*y) + x) - x*y) assuming offset = 0. - Peter Luschny, Oct 21 2022

Example

The triangle T(n, m)  begins:
n\m 0     1      2        3        4         5         6          7       8
1:  1
2: -1     2
3:  1    -8      9
4: -1    24    -81       64
5:  1   -64    486    -1024      625
6: -1   160  -2430    10240   -15625      7776
7:  1  -384  10935   -81920   234375   -279936    117649
8: -1   896 -45927   573440 -2734375   5878656  -5764801    2097152
9:  1 -2048 183708 -3670016 27343750 -94058496 161414428 -134217728 4304672
...
n = 10: -1 4608 -708588 22020096 -246093750 1269789696 -3389702988 4831838208 -3486784401 1000000000. [Reformatted by Wolfdieter Lang, Oct 12 2022]
-----------------------------------------------------------------------------
p(2,x) = -1+2*x = (1/(2*x))*x*(d/dx)*x*(d/dx)*(x-1)^2.

Maple

# Assuming offset 0.
seq(seq((-1)^(n-k)*binomial(n, k)*(k+1)^n, k=0..n), n=0..8);
# Alternative:
egf := x -> 1/(exp(LambertW(-exp(-x)*x*y) + x) - x*y):
ser := x -> series(egf(x), x, 12):
row := n -> seq(coeff(n!*coeff(ser(x), x, n), y, k), k=0..n):
seq(print(row(n)), n = 0..8); # Peter Luschny, Oct 21 2022

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, A258773.

Sequence in context: A188922 A036296 A078105 * A284211 A246403 A355183

Adjacent sequences: A075510 A075511 A075512 * A075514 A075515 A075516

Keyword

sign,tabl,easy

Author

Wolfdieter Lang, Oct 02 2002

Status

approved