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A039683
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Signed double Pochhammer triangle: expansion of x(x-2)(x-4)..(x-2n+2).
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21
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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
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OFFSET
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1,2
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COMMENTS
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T(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
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
The signed triangle is also the inverse Bell transform of A000079 (see Luschny link). - John Keith, Nov 24 2020
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LINKS
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FORMULA
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T(n, m) = T(n-1, m-1) - 2*(n-1)*T(n-1, m), n >= m >= 1; T(n, m) := 0, n<m; T(n, 0) := 0, T(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
T(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
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EXAMPLE
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Triangle starts:
{1},
{2,1},
{8,6,1},
{48,44,12,1},
...
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)
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MATHEMATICA
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Table[ Rest@ CoefficientList[ Product[ z-k, {k, 0, 2p-2, 2} ], z ], {p, 6} ]
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PROG
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(Sage) # uses[bell_transform from A264428]
# Unsigned values and an additional first column (1, 0, 0, ...).
dblfact = a.list(n)
return bell_transform(n, dblfact)
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CROSSREFS
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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).
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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