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%I #37 Jul 21 2022 13:47:59
%S 1,0,1,0,3,1,0,13,9,1,0,75,79,18,1,0,541,765,265,30,1,0,4683,8311,
%T 3870,665,45,1,0,47293,100989,59101,13650,1400,63,1,0,545835,1362439,
%U 960498,278901,38430,2618,84,1,0,7087261,20246445,16700545,5844510,1012431,92610,4494,108,1
%N Exponential Riordan array [1, 1/(2-e^x)-1].
%C This is also the matrix product of the Stirling set numbers and the unsigned Lah numbers.
%C This is also the Bell transform of A000670(n+1). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 29 2016
%F Row sums are given by A075729.
%F T(n,1) = A000670(n) for n>=1.
%F T(n,k) = n!/k! * [x^n] (1/(2-exp(x))-1)^k. - _Alois P. Heinz_, Apr 17 2015
%e Number triangle starts:
%e 1;
%e 0, 1;
%e 0, 3, 1;
%e 0, 13, 9, 1;
%e 0, 75, 79, 18, 1;
%e 0, 541, 765, 265, 30, 1;
%e ...
%p T:= (n, k)-> n!*coeff(series((1/(2-exp(x))-1)^k/k!, x, n+1), x, n):
%p seq(seq(T(n, k), k=0..n), n=0..10); # _Alois P. Heinz_, Apr 17 2015
%p # The function BellMatrix is defined in A264428.
%p BellMatrix(n -> polylog(-n-1, 1/2)/2, 9); # _Peter Luschny_, Jan 29 2016
%t T[n_, k_] := n!*SeriesCoefficient[(1/(2 - Exp[x]) - 1)^k/k!, {x, 0, n}];
%t Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, May 23 2016, after _Alois P. Heinz_ *)
%t (* The function BellMatrix is defined in A264428. *)
%t BellMatrix[PolyLog[-#-1, 1/2]/2&, 9] (* _Jean-François Alcover_, May 23 2016, after _Peter Luschny_ *)
%t RiordanArray[d_, h_, n_] := RiordanArray[d, h, n, False];
%t RiordanArray[d_Function|d_Symbol, h_Function|h_Symbol, n_, exp_:(True | False)] := Module[{M, td, th, k, m},
%t M[_, _] = 0;
%t td = PadRight[CoefficientList[d[x] + O[x]^n, x], n];
%t th = PadRight[CoefficientList[h[x] + O[x]^n, x], n];
%t For[k = 0, k <= n - 1, k++, M[k, 0] = td[[k + 1]]];
%t For[k = 1, k <= n - 1, k++,
%t For[m = k, m <= n - 1, m++,
%t M[m, k] = Sum[M[j, k - 1]*th[[m - j + 1]], {j, k - 1, m - 1}]]];
%t If[exp,
%t u = 1;
%t For[k = 1, k <= n - 1, k++,
%t u *= k;
%t For[m = 0, m <= k, m++,
%t j = If[m == 0, u, j/m];
%t M[k, m] *= j]]];
%t Table[M[m, k], {m, 0, n - 1}, {k, 0, m}]];
%t RiordanArray[1&, 1/(2 - Exp[#])-1&, 10, True] // Flatten (* _Jean-François Alcover_, Jul 16 2019, after Sage program *)
%o (Sage)
%o def riordan_array(d, h, n, exp=false):
%o def taylor_list(f,n):
%o t = SR(f).taylor(x,0,n-1).list()
%o return t + [0]*(n-len(t))
%o td = taylor_list(d,n)
%o th = taylor_list(h,n)
%o M = matrix(QQ,n,n)
%o for k in (0..n-1): M[k,0] = td[k]
%o for k in (1..n-1):
%o for m in (k..n-1):
%o M[m,k] = add(M[j,k-1]*th[m-j] for j in (k-1..m-1))
%o if exp:
%o u = 1
%o for k in (1..n-1):
%o u *= k
%o for m in (0..k):
%o j = u if m==0 else j/m
%o M[k,m] *= j
%o return M
%o riordan_array(1, 1/(2-exp(x)) - 1, 8, exp=true)
%o # As a matrix product:
%o def Lah(n, k):
%o if n == k: return 1
%o if k<0 or k>n: return 0
%o return (k*n*gamma(n)^2)/(gamma(k+1)^2*gamma(n-k+1))
%o matrix(ZZ, 8, stirling_number2)*matrix(ZZ, 8, Lah)
%Y Cf. A088729 which is a variant based on an (1,1)-offset of the number triangles.
%Y Cf. A131222 which is the matrix product of the unsigned Lah numbers and the Stirling cycle numbers.
%Y Cf. A000670, A075729.
%K nonn,tabl
%O 0,5
%A _Peter Luschny_, Apr 17 2015
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