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 A062715 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
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET -1,5 COMMENTS 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. From Peter Bala, Sep 08 2011: (Start) 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) From Peter Bala, Oct 13 2011: (Start) 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) REFERENCES 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. LINKS Reinhard Zumkeller, Rows n = -1..120 of triangle, flattened EXAMPLE 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. MATHEMATICA 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 *) PROG (Haskell) a062715 n k = a062715_tabl !! n !! k a062715_row n = a062715_tabl !! n a062715_tabl = 1 : zipWith (:) a000012_list a038207_tabl -- Reinhard Zumkeller, Feb 27 2013 CROSSREFS Cf. A038207 (left column dropped). Cf. A001896, A001897, A007318, A060187, A132440, A135225, A145901. Sequence in context: A202979 A156006 A137854 * A100631 A154867 A064298 Adjacent sequences:  A062712 A062713 A062714 * A062716 A062717 A062718 KEYWORD nonn,tabl,easy,nice AUTHOR STATUS approved

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Last modified December 18 18:54 EST 2018. Contains 318243 sequences. (Running on oeis4.)