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 A256893 Exponential Riordan array [1, 1/(2-e^x)]. 66
 1, 0, 1, 0, 3, 1, 0, 13, 9, 1, 0, 75, 79, 18, 1, 0, 541, 765, 265, 30, 1, 0, 4683, 8311, 3870, 665, 45, 1, 0, 47293, 100989, 59101, 13650, 1400, 63, 1, 0, 545835, 1362439, 960498, 278901, 38430, 2618, 84, 1, 0, 7087261, 20246445, 16700545, 5844510, 1012431, 92610, 4494, 108, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS This is also the matrix product of the Stirling set numbers and the unsigned Lah numbers. This is also the Bell transform of A000670(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 29 2016 LINKS FORMULA Row sums are given by A075729. T(n,1) = A000670(n) for n>=1. T(n,k) = n!/k! * [x^n] (1/(2-exp(x))-1)^k. - Alois P. Heinz, Apr 17 2015 EXAMPLE Number triangle starts:   1;   0,    1;   0,    3,     1;   0,   13,     9,     1;   0,   75,    79,    18,    1;   0,  541,   765,   265,   30,   1;   ... MAPLE T:= (n, k)-> n!*coeff(series((1/(2-exp(x))-1)^k/k!, x, n+1), x, n): seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Apr 17 2015 # The function BellMatrix is defined in A264428. BellMatrix(n -> polylog(-n-1, 1/2)/2, 9); # Peter Luschny, Jan 29 2016 MATHEMATICA T[n_, k_] := n!*SeriesCoefficient[(1/(2 - Exp[x]) - 1)^k/k!, {x, 0, n}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 23 2016, after Alois P. Heinz *) (* The function BellMatrix is defined in A264428. *) BellMatrix[PolyLog[-#-1, 1/2]/2&, 9] (* Jean-François Alcover, May 23 2016, after Peter Luschny *) RiordanArray[d_, h_, n_] := RiordanArray[d, h, n, False]; RiordanArray[d_Function|d_Symbol, h_Function|h_Symbol, n_, exp_:(True | False)] := Module[{M, td, th, k, m}, M[_, _] = 0; td = PadRight[CoefficientList[d[x] + O[x]^n, x], n]; th = PadRight[CoefficientList[h[x] + O[x]^n, x], n]; For[k = 0, k <= n - 1, k++, M[k, 0] = td[[k + 1]]]; For[k = 1, k <= n - 1, k++,   For[m = k, m <= n - 1, m++,     M[m, k] = Sum[M[j, k - 1]*th[[m - j + 1]], {j, k - 1, m - 1}]]]; If[exp,   u = 1;   For[k = 1, k <= n - 1, k++,     u *= k;     For[m = 0, m <= k, m++,       j = If[m == 0, u, j/m];       M[k, m] *= j]]]; Table[M[m, k], {m, 0, n - 1}, {k, 0, m}]]; RiordanArray[1&, 1/(2 - Exp[#])&, 10, True] // Flatten (* Jean-François Alcover, Jul 16 2019, after Sage program *) PROG (Sage) def riordan_array(d, h, n, exp=false):     def taylor_list(f, n):         t = SR(f).taylor(x, 0, n-1).list()         return t + [0]*(n-len(t))     td = taylor_list(d, n)     th = taylor_list(h, n)     M = matrix(QQ, n, n)     for k in (0..n-1): M[k, 0] = td[k]     for k in (1..n-1):         for m in (k..n-1):             M[m, k] = add(M[j, k-1]*th[m-j] for j in (k-1..m-1))     if exp:         u = 1         for k in (1..n-1):             u *= k             for m in (0..k):                 j = u if m==0 else j/m                 M[k, m] *= j     return M riordan_array(1, 1/(2-exp(x)), 8, exp=true) # As a matrix product: def Lah(n, k):     if n == k: return 1     if k<0 or  k>n: return 0     return (k*n*gamma(n)^2)/(gamma(k+1)^2*gamma(n-k+1)) matrix(ZZ, 8, stirling_number2)*matrix(ZZ, 8, Lah) CROSSREFS Cf. A088729 which is a variant based on an (1,1)-offset of the number triangles. Cf. A131222 which is the matrix product of the unsigned Lah numbers and the Stirling cycle numbers. Cf. A000670, A075729. Sequence in context: A271704 A307419 A256892 * A137431 A131222 A228334 Adjacent sequences:  A256890 A256891 A256892 * A256894 A256895 A256896 KEYWORD nonn,tabl AUTHOR Peter Luschny, Apr 17 2015 STATUS approved

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Last modified May 28 15:33 EDT 2020. Contains 334684 sequences. (Running on oeis4.)