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A062715
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Triangle T(i,j) (i >= -1, -1<=j<=i) whose (i,j)-th entry is 1 if j=-1 otherwise binomial(i,j)*2^(i-j).
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8
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1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 12, 6, 1, 1, 16, 32, 24, 8, 1, 1, 32, 80, 80, 40, 10, 1, 1, 64, 192, 240, 160, 60, 12, 1, 1, 128, 448, 672, 560, 280, 84, 14, 1, 1, 256, 1024, 1792, 1792, 1120, 448, 112, 16, 1, 1, 512, 2304, 4608, 5376, 4032, 2016, 672, 144, 18, 1, 1, 1024, 5120, 11520, 15360, 13440
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OFFSET
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-1,5
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COMMENTS
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As an upper right triangle, table rows give numbers of empty sets, points, edges, faces, cubes, 4D hypercubes etc. in hypercubes of increasing dimension by column.
Let P denote Pascal's triangle A007318. This array, call it M, is obtained by augmenting P^2 with an initial column of 1's on the left.
The array has connections with the Bernoulli polynomials B(n,x). The infinitesimal generator S of M is the array such that exp(S) = M. The array S is obtained by augmenting the infinitesimal generator 2*A132440 of P^2 with an initial column [0,1,0,-1/3,0,7/15,...] on the left. The entries in this column, apart from the initial zero, are the cosecant numbers 2^n*B(n,1/2), n>=1. See A001896 and A001897.
The array M is also connected with the problem of summing powers of consecutive odd integers. In the array M^n, the entry in position p+1 of the first column is equal to sum {k = 1..n} (2*k-1)^p - see the Example section below.
For similar results when the array P, rather than P^2, is augmented with a column of 1's see A135225.
(End)
The first column of the array (I-t*A062715)^-1, I the identity, begins [1/(1-t), t/(1-t)^2, t*(1+t)/(1-t)^3, t*(1+6*t+t^2)/(1-t)^4, ...]. The numerator polynomials, apart from the initial 1, are the type B Eulerian polynomials of A060187 (but with an extra factor of t).
The polynomials in t in the first column of the array ((1+t)*I-t*A062715)^-1 are, apart from the initial 1, the row polynomials of A145901 (but again with an extra factor of t). (End)
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REFERENCES
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B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.
S. J. Cyvin et al., Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.
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LINKS
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EXAMPLE
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1; 1,1; 1,2,1; 1,4,4,1; 1,8,12,6,1; ...
The infinitesimal generator for M begins
/0
|1........0
|0........2...0
|-1/3.....0...4....0
|0........0...0....6....0
|7/15.....0...0....0....8....0
|0........0...0....0....0...10....0
|-31/21...0...0....0....0....0...12....0
|...
\
The array M^n begins
/1
|1+1+ ... +1..............1
|1+3+...+(2*n-1)........2*n...........1
|1+3^2+...+(2*n-1)^2...(2*n)^2...2*(2*n)..........1
|1+3^3+...+(2*n-1)^3...(2*n)^3...3*(2*n)^2...3*(2*n)...1
|...
\
More generally, the array M^t, defined as exp(t*S) for complex t, begins
/1
|Q(1,t)......1
|Q(2,t)....2*t..........1
|Q(3,t)....4*t^2......4*t.......1
|Q(4,t)....8*t^3.....12*t^2...6*t.....1
|...
\
where Q(n,t) := 2^(n-1)/n*(B(n,t+1/2)-B(n,1/2)) satisfies Q(n,t+1)-Q(n,t) = (2*t+1)^(n-1) for n>=1.
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MATHEMATICA
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imax = 10; t[i_, -1] = 1; t[i_, j_] := Binomial[i, j]*2^(i-j); Flatten[ Table[t[i, j], {i, -1, imax}, {j, -1, i}]] (* Jean-François Alcover, Nov 23 2011 *)
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PROG
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(Haskell)
a062715 n k = a062715_tabl !! n !! k
a062715_row n = a062715_tabl !! n
a062715_tabl = 1 : zipWith (:) a000012_list a038207_tabl
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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