A025966 -id:A025966 - cp's OEIS frontend

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

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)

A016283 a(n) = 6^n/8 - 4^(n-1) + 2^(n-3).

Original entry on oeis.org

0, 0, 1, 12, 100, 720, 4816, 30912, 193600, 1194240, 7296256, 44301312, 267904000, 1615810560, 9728413696, 58504691712, 351565004800, 2111537479680, 12677814747136, 76101248090112, 456744927232000

Offset: 0

N. J. A. Sloane

nonn

Number of rectangles that can be formed from the vertices of an n-dimensional cube. E.g., a(3)=12 because the three-dimensional cube has six faces plus six rectangles passing through the center of the cube. Cf. A064436: each rectangle on the cube provides an opportunity for a function not to be a linear threshold function, by alternating in value around the rectangle. - Matthew Cook, Jan 26 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..150
Index entries for linear recurrences with constant coefficients, signature (12, -44, 48).

Third column of triangle A075497.

Cf. A025966.

[6^n/8 - 4^(n-1) + 2^(n-3): n in [0..25]]; // Vincenzo Librandi, Apr 26 2011

[seq(9/2*6^n-4*4^n+1/2*2^n,n=0..20)]; # Detlef Pauly (dettodet(AT)yahoo.de), Dec 04 2001

CoefficientList[Series[x^2/((1 - 2 x) (1 - 4 x) (1 - 6 x)), {x, 0, 20}], x] (* Michael De Vlieger, Jan 31 2018 *)

[((6^n - 2^n)/4-(4^n - 2^n)/2)/2 for n in range(0,21)] # Zerinvary Lajos, Jun 05 2009

a(n) = (2^n)*Stirling2(n+3, 3), n >= 0, with Stirling2(n, m) = A008277(n, m).
G.f.: x^2/((1-2*x)*(1-4*x)*(1-6*x)).
E.g.f.: (exp(2*x) - 8*exp(4*x) + 9*exp(6*x))/2!.
a(n) =((6^n - 2^n)/4 - (4^n - 2^n)/2)/2 , n >= 0. - Zerinvary Lajos, Jun 05 2009

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

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A016283 a(n) = 6^n/8 - 4^(n-1) + 2^(n-3).

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A075510 Fifth column of triangle A075497.

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