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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).
8

%I #26 Jul 25 2017 04:12:28

%S 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,

%T 64,192,240,160,60,12,1,1,128,448,672,560,280,84,14,1,1,256,1024,1792,

%U 1792,1120,448,112,16,1,1,512,2304,4608,5376,4032,2016,672,144,18,1,1,1024,5120,11520,15360,13440

%N 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).

%C 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.

%C From _Peter Bala_, Sep 08 2011: (Start)

%C 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.

%C 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.

%C 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.

%C For similar results when the array P, rather than P^2, is augmented with a column of 1's see A135225.

%C (End)

%C From _Peter Bala_, Oct 13 2011: (Start)

%C 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).

%C 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)

%D B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.

%D S. J. Cyvin et al., Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.

%H Reinhard Zumkeller, <a href="/A062715/b062715.txt">Rows n = -1..120 of triangle, flattened</a>

%e 1; 1,1; 1,2,1; 1,4,4,1; 1,8,12,6,1; ...

%e The infinitesimal generator for M begins

%e /0

%e |1........0

%e |0........2...0

%e |-1/3.....0...4....0

%e |0........0...0....6....0

%e |7/15.....0...0....0....8....0

%e |0........0...0....0....0...10....0

%e |-31/21...0...0....0....0....0...12....0

%e |...

%e \

%e The array M^n begins

%e /1

%e |1+1+ ... +1..............1

%e |1+3+...+(2*n-1)........2*n...........1

%e |1+3^2+...+(2*n-1)^2...(2*n)^2...2*(2*n)..........1

%e |1+3^3+...+(2*n-1)^3...(2*n)^3...3*(2*n)^2...3*(2*n)...1

%e |...

%e \

%e More generally, the array M^t, defined as exp(t*S) for complex t, begins

%e /1

%e |Q(1,t)......1

%e |Q(2,t)....2*t..........1

%e |Q(3,t)....4*t^2......4*t.......1

%e |Q(4,t)....8*t^3.....12*t^2...6*t.....1

%e |...

%e \

%e 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.

%t 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 *)

%o (Haskell)

%o a062715 n k = a062715_tabl !! n !! k

%o a062715_row n = a062715_tabl !! n

%o a062715_tabl = 1 : zipWith (:) a000012_list a038207_tabl

%o -- _Reinhard Zumkeller_, Feb 27 2013

%Y Cf. A038207 (left column dropped).

%Y Cf. A001896, A001897, A007318, A060187, A132440, A135225, A145901.

%K nonn,tabl,easy,nice

%O -1,5

%A _N. J. A. Sloane_