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 A200139 Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. 8
 1, 1, 1, 2, 3, 1, 4, 8, 5, 1, 8, 20, 18, 7, 1, 16, 48, 56, 32, 9, 1, 32, 112, 160, 120, 50, 11, 1, 64, 256, 432, 400, 220, 72, 13, 1, 128, 576, 1120, 1232, 840, 364, 98, 15, 1, 256, 1280, 2816, 3584, 2912, 1568, 560, 128, 17, 1, 512, 2816, 6912, 9984, 9408, 6048, 2688, 816, 162, 19, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Riordan array ((1-x)/(1-2x),x/(1-2x)). Product A097805*A007318 as infinite lower triangular arrays. Product A193723*A130595 as infinite lower triangular arrays. T(n,k) is the number of ways to place n unlabeled objects into any number of labeled bins (with at least one object in each bin) and then designate k of the bins. - Geoffrey Critzer, Nov 18 2012 Apparently, rows of this array are unsigned diagonals of A028297. - Tom Copeland, Oct 11 2014 Unsigned A118800, so my conjecture above is true. - Tom Copeland, Nov 14 2016 LINKS FORMULA T(n,k) = 2*T(n-1,k)+T(n-1,k-1) with T(0,0)=T(1,0)=T(1,1)=1 and T(n,k)=0 for k<0 or for n=0 T(n-1-j,k-1)*2^j. T = A007318*A059260, so the row polynomials of this entry are given umbrally by p_n(x) = (1 + q.(x))^n, where q_n(x) are the row polynomials of A059260 and (q.(x))^k = q_k(x). Consequently, the e.g.f. is exp[tp.(x)] = exp[t(1+q.(x))] = e^t exp(tq.(x)) = [1 + (x+1)e^((x+2)t)]/(x+2), and p_n(x) = (x+1)(x+2)^(n-1) for n > 0. - Tom Copeland, Nov 15 2016 T^(-1) = A130595*(padded A130595), differently signed A118801. Cf. A097805. - Tom Copeland, Nov 17 2016 The n-th row polynomial in descending powers of x is the n-th Taylor polynomial of the rational function (1 + x)/(1 + 2*x) * (1 + 2*x)^n about 0. For example, for n = 4, (1 + x)/(1 + 2*x) * (1 + 2*x)^4 = (8*x^4 + 20*x*3 + 18*x^2 + 7*x + 1) + O(x^5). - Peter Bala, Feb 24 2018 EXAMPLE Triangle begins:    1    1,   1    2,   3,   1    4,   8,   5,   1    8,  20,  18,   7,   1   16,  48,  56,  32,   9,   1   32, 112, 160, 120,  50,  11,   1 MATHEMATICA nn=15; f[list_]:=Select[list, #>0&]; Map[f, CoefficientList[Series[(1-x)/(1-2x-y x) , {x, 0, nn}], {x, y}]]//Grid  (* Geoffrey Critzer, Nov 18 2012 *) CROSSREFS Cf. A118800 (signed version), A081277, A039991, A001333 (antidiagonal sums), A025192 (row sums); diagonals: A000012, A005408, A001105, A002492, A072819l; columns: A011782, A001792, A001793, A001794, A006974, A006975, A006976. Cf. A007318, A028297, A059260, A097805, A118801, A130595. Sequence in context: A179738 A187889 A118800 * A321621 A321629 A075297 Adjacent sequences:  A200136 A200137 A200138 * A200140 A200141 A200142 KEYWORD nonn,tabl,easy AUTHOR Philippe Deléham, Nov 13 2011 STATUS approved

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Last modified October 22 19:53 EDT 2019. Contains 328319 sequences. (Running on oeis4.)