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A075497 Stirling2 triangle with scaled diagonals (powers of 2). 16
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

P. Bala, The white diamond product of power series

Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.

J. R. Britnell, M. 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.

W. Lang, First 10 rows.

T. Mansour, Generalization of some identities involving the Fibonacci numbers, arXiv:math/0301157 [math.CO], 2003.

FORMULA

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)

EXAMPLE

[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

MAPLE

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

MATHEMATICA

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

PROG

(Sage)

# The function inverse_bell_transform is defined in 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

(PARI)

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

CROSSREFS

Columns 1-7 are A000079, A006516, A016283, A025966, A075510-A075512.

Row sums are A004211.

Cf. A008277, A075498-A075505.

Sequence in context: A114192 A114656 A294440 * A158983 A261642 A185947

Adjacent sequences:  A075494 A075495 A075496 * A075498 A075499 A075500

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Oct 02 2002

STATUS

approved

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Last modified February 23 02:29 EST 2019. Contains 320411 sequences. (Running on oeis4.)