A075512 -id:A075512 - cp's OEIS frontend

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

Wolfdieter Lang, Oct 02 2002

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

			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
  [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.

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

Wolfdieter Lang, On a Certain Family of Sidi Polynomials, May 2023.
Harlan J. Brothers, Pascal's triangle, Sidi polynomials, and powers of e, Missouri J. Math. Sci. (2025) Vol. 37, No. 1, 67-78.
Doron S. Lubinsky and Herbert 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.

Cf. A075510, A075511, A075512, A074932, A075515, A075516, A075906..A075925, A076002..A076013, A258773.

# 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

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 *)

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

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

A075497 Stirling2 triangle with scaled diagonals (powers of 2).

Original entry on oeis.org

1, 2, 1, 4, 6, 1, 8, 28, 12, 1, 16, 120, 100, 20, 1, 32, 496, 720, 260, 30, 1, 64, 2016, 4816, 2800, 560, 42, 1, 128, 8128, 30912, 27216, 8400, 1064, 56, 1, 256, 32640, 193600, 248640, 111216, 21168, 1848, 72, 1

Offset: 1

Wolfdieter Lang, Oct 02 2002

This is a lower triangular infinite matrix of the Jabotinsky type. See the D. E. Knuth reference given in A039692 for exponential convolution arrays.
The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(2*z) - 1)*x/2) - 1.
Subtriangle of (0, 2, 0, 4, 0, 6, 0, 8, 0, 10, 0, 12, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 13 2013
Also the inverse Bell transform of the double factorial of even numbers Product_ {k=0..n-1} (2*k+2) (A000165). For the definition of the Bell transform see A264428 and for cross-references A265604. - Peter Luschny, Dec 31 2015
This is the exponential Riordan array [exp(2*x), (exp(2*x) - 1)/2] belonging to the derivative subgroup of the exponential Riordan group. In the notation of Corcino, this is the triangle of (2, 2)-Stirling numbers of the second kind. A factorization of the array as an infinite product is given in the example section. - Peter Bala, Feb 20 2025

			Triangle begins:
  [1];
  [2,1];
  [4,6,1]; p(3,x) = x*(4 + 6*x + x^2).
  ...;
Triangle (0, 2, 0, 4, 0, 6, 0, 8, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, ...) begins:
  1
  0,  1
  0,  2,   1
  0,  4,   6,   1
  0,  8,  28,  12,  1
  0, 16, 120, 100, 20, 1. - _Philippe Deléham_, Feb 13 2013
From _Peter Bala_, Feb 23 2025: (Start)
The array factorizes as
/ 1               \       /1             \ /1             \ /1            \
| 2    1           |     | 2   1          ||0  1           ||0  1          |
| 4    6   1       |  =  | 4   4   1      ||0  2   1       ||0  0  1       | ...
| 8   28  12   1   |     | 8  12   6  1   ||0  4   4  1    ||0  0  2  1    |
|16  120 100  20  1|     |16  32  24  8  1||0  8  12  6  1 ||0  0  4  4  1 |
|...               |     |...             ||...            ||...           |
where, in the infinite product on the right-hand side, the first array is the Riordan array (1/(1 - 2*x), x/(1 - 2*x)) = P^2, where P denotes Pascal's triangle. See A038207. Cf. A143494. (End)

Alois P. Heinz, Rows n = 1..141, flattened
Peter Bala, The white diamond product of power series
Peter Bala, Factorising (r,b)-Stirling arrays
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
John R. Britnell and Mark Wildon, Bell numbers, partition moves and the eigenvalues of the random-to-top shuffle in types A, B and D, arXiv 1507.04803 [math.CO], 2015.
Roberto B. Corcino, The (r, β)-Stirling Numbers, The Mindanao Forum, Vol. XIV, No.2, pp. 91-99, 1999.
Roberto B. Corcino and Maribeth B. Montero, The (r, β)-Stirling Numbers in the Context of 0-1 Tableau, Jour. Math. Soc. of the Philippines, ISSN 0115-6926, Vol. 32, No. 1 (2009), pp. 45-52
Paweł Hitczenko, A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality, arXiv:2403.03422 [math.CO], 2024. See p. 8.
Wolfdieter Lang, First 10 rows.
Toufik Mansour, Generalization of some identities involving the Fibonacci numbers, arXiv:math/0301157 [math.CO], 2003.
Emanuele Munarini, Characteristic, admittance and matching polynomials of an antiregular graph, Appl. Anal. Discrete Math 3 (1) (2009) 157-176.

Columns 1-7 are A000079, A006516, A016283, A025966, A075510-A075512.
Row sums are A004211.
Cf. A008277, A075498-A075505.

with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
       `if`(i<1, 0, add(x^j*multinomial(n, n-i*j, i$j)/j!*add(
        binomial(i, 2*k), k=0..i/2)^j*b(n-i*j, i-1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):
seq(T(n), n=1..12);  # Alois P. Heinz, Aug 13 2015
# Alternatively, giving the triangle in the form displayed in the Example section:
gf := exp(x*exp(z)*sinh(z)):
X := n -> series(gf, z, n+2):
Z := n -> n!*expand(simplify(coeff(X(n), z, n))):
A075497_row := n -> op(PolynomialTools:-CoefficientList(Z(n), x)):
seq(A075497_row(n), n=0..9); # Peter Luschny, Jan 14 2018

Table[(2^(n - m)) StirlingS2[n, m], {n, 9}, {m, n}] // Flatten (* Michael De Vlieger, Dec 31 2015 *)

for(n=1, 11, for(m=1, n, print1(2^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

# uses[inverse_bell_transform from A265605]
multifact_2_2 = lambda n: prod(2*k + 2 for k in (0..n-1))
inverse_bell_matrix(multifact_2_2, 9) # Peter Luschny, Dec 31 2015

a(n, m) = (2^(n-m)) * Stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*2)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 2*m*a(n-1, m) + a(n-1, m-1), n >= m >= 1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-2*k*x), m >= 1.
E.g.f. for m-th column: (((exp(2*x)-1)/2)^m)/m!, m >= 1.
The row polynomials in t are given by D^n(exp(x*t)) evaluated at x = 0, where D is the operator (1+2*x)*d/dx. Cf. A008277. - Peter Bala, Nov 25 2011
From Peter Bala, Jan 13 2018: (Start)
n-th row polynomial R(n,x)= x o x o ... o x (n factors), where o is the deformed Hadamard product of power series defined in Bala, section 3.1.
R(n+1,x)/x = (x + 2) o (x + 2) o...o (x + 2) (n factors).
R(n+1,x) = x*Sum_{k = 0..n} binomial(n,k)*2^(n-k)*R(k,x).
Dobinski-type formulas: R(n,x) = exp(-x/2)*Sum_{i >= 0} (2*i)^n* (x/2)^i/i!; 1/x*R(n+1,x) = exp(-x/2)*Sum_{i >= 0} (2 + 2*i)^n* (x/2)^i/i!. (End)

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

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A075497 Stirling2 triangle with scaled diagonals (powers of 2).

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A075511 Sixth column of triangle A075497.

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