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A039683 Signed double Pochhammer triangle: expansion of x(x-2)(x-4)..(x-2n+2). 21
1, -2, 1, 8, -6, 1, -48, 44, -12, 1, 384, -400, 140, -20, 1, -3840, 4384, -1800, 340, -30, 1, 46080, -56448, 25984, -5880, 700, -42, 1, -645120, 836352, -420224, 108304, -15680, 1288, -56, 1, 10321920, -14026752, 7559936, -2153088, 359184, -36288, 2184, -72, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n,m) = R_n^m(a=0,b=2) in the notation of the given reference.

Exponential Riordan array [1/(1+2x),log(1+2x)/2]. The unsigned triangle is [1/(1-2x),log(1/sqrt(1-2x))]. - Paul Barry_, Apr 29 2009

The n-th row is related to the expansion of z^(-2n)*(z^3 d/dz)^n in polynomials of the Euler operator D=(z d/dz). E.g., z^(-6)(z^3 d/dz)^3 = D^3 + 6 D^2 + 8 D. See Copeland link for relations to Bell / Exponential / Touchard polynomial operators. - Tom Copeland, Nov 14 2013

A refinement of this array is given by A231846. - Tom Copeland, Nov 15 2013

Also the Bell transform of the double factorial of even numbers A000165 except that the values are unsigned and in addition a first column (1,0,0 ...) is added on the left side of the triangle. For the Bell transform of the double factorial of odd numbers A001147 see A132062. For the definition of the Bell transform see A264428. - Peter Luschny, Dec 20 2015

LINKS

Table of n, a(n) for n=1..45.

Richell O. Celeste, Roberto B. Corcino, Ken Joffaniel M. Gonzales. Two Approaches to Normal Order Coefficients. Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.

Tom Copeland, Addendum to Mathemagical Forests.

P. Feijão, F. V. Martinez, A. Thévenin, On the distribution of cycles and paths in multichromosomal breakpoint graphs and the expected value of rearrangement distance, BMC Bioinformatics 16:Suppl19 (2015), S1. doi:10.1186/1471-2105-16-S19-S1

Lisa Glaser, Causal set actions in various dimensions, J. Phys.: Conf. Ser. 306 (2011), 012041.

W. Lang, First 9 rows and comment.

D. S. Mitrinovic, M. S. Mitrinovic, Tableaux d'une classe de nombres relies aux nombres de Stirling, Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. 77 (1962).

FORMULA

a(n, m) = a(n-1, m-1) - 2*(n-1)*a(n-1, m), n >= m >= 1; a(n, m) := 0, n<m; a(n, 0) := 0, a(1, 1)=1. E.g.f. for m-th column of signed triangle: (((log(1+2*x))/2)^m)/m!.

E.g.f.: (1+2*x)^(y/2). O.g.f. for n-th row of signed triangle: Sum_{m=0..n} stirling1(n, m)*2^(n-m)*x^m. - Vladeta Jovovic, Feb 11 2003

a(n, m) = S1(n, m)*2^(n-m), with S1(n, m) := A008275(n, m) (signed Stirling1 triangle).

The production matrix below is A038207 with the first row removed. With the initial index n = 0, the associated differential raising operator is R = e^(2D)*x = (2+x)*e^(2D) with D = d/dx, i.e., R p_n(x) = p_(n+1)(x) where p_n(x) is the n-th unsigned row polynomial and p_0(x) = 1, so p_(n+1)(x) = (2+x) * p_n(2+x). - Tom Copeland, Oct 11 2016

EXAMPLE

Triangle starts:

{1},

{2,1},

{8,6,1},

{48,44,12,1},

...

From Paul Barry, Apr 29 2009: (Start)

The unsigned triangle [1/(1-2x),log(1/sqrt(1-2x))] has production matrix

2, 1,

4, 4, 1,

8, 12, 6, 1,

16, 32, 24, 8, 1,

32, 80, 80, 40, 10, 1,

64, 192, 240, 160, 60, 12, 1

which is A007318^{2} beheaded. (End)

MATHEMATICA

Table[ Rest@ CoefficientList[ Product[ z-k, {k, 0, 2p-2, 2} ], z ], {p, 6} ]

PROG

(Sage)

# Unsigned values and an additional first column (1, 0, 0, ...).

# The function bell_transform is defined in A264428.

def A039683_unsigned_row(n):

    a = sloane.A000165

    dblfact = a.list(n)

    return bell_transform(n, dblfact)

[A039683_unsigned_row(n) for n in (0..9)] # Peter Luschny, Dec 20 2015

CROSSREFS

First column (unsigned triangle) is (2(n-1))!! = 1, 2, 8, 48, 384...= A000165(n-1) and the row sums (unsigned) are (2n-1)!! = 1, 3, 15, 105, 945... = A001147(n-1).

Cf. A051141, A051142.

Cf. A000165, A132062, A264428.

Cf. A038207.

Sequence in context: A193735 A114193 A231846 * A108084 A108085 A272983

Adjacent sequences:  A039680 A039681 A039682 * A039684 A039685 A039686

KEYWORD

sign,tabl

AUTHOR

Wouter Meeussen

EXTENSIONS

Additional comments from Wolfdieter Lang

Title revised by Tom Copeland, Dec 21 2013

STATUS

approved

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Last modified February 25 13:30 EST 2018. Contains 299654 sequences. (Running on oeis4.)