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A131222 Exponential Riordan array [1, log((1-x)/(1-2x))].

%I A131222 #43 Jan 09 2024 12:18:46
%S A131222 1,0,1,0,3,1,0,14,9,1,0,90,83,18,1,0,744,870,275,30,1,0,7560,10474,
%T A131222 4275,685,45,1,0,91440,143892,70924,14805,1435,63,1,0,1285200,2233356,
%U A131222 1274196,324289,41160,2674,84,1
%N A131222 Exponential Riordan array [1, log((1-x)/(1-2x))].
%C A131222 This is also the matrix product of the unsigned Lah numbers and the Stirling cycle numbers. See also A079639 and A079640 for variants based on an (1,1)-offset of the number triangles. - _Peter Luschny_, Apr 12 2015
%C A131222 The Bell transform of A029767(n+1). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 18 2016
%C A131222 Essentially the same as A079638. - _Peter Bala_, Feb 12 2022
%H A131222 Alois P. Heinz, <a href="/A131222/b131222.txt">Rows n = 0..140, flattened</a>
%H A131222 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barry4/barry122.html">Exponential Riordan arrays and permutation enumeration</a>,Journal of Integer Sequences, Vol. 13 (2010).
%F A131222 Row sums are A002866.
%F A131222 Second column is A029767.
%F A131222 T(n,m) = n! * Sum_{k=m..n} Stirling1(k,m)*2^(n-k)*binomial(n-1,k-1)/k!, n >= m >= 0. - _Vladimir Kruchinin_, Sep 27 2012
%e A131222 Number triangle starts:
%e A131222   1,
%e A131222   0,   1;
%e A131222   0,   3,   1;
%e A131222   0,  14,   9,   1;
%e A131222   0,  90,  83,  18,  1;
%e A131222   0, 744, 870, 275, 30,  1;
%e A131222   ...
%p A131222 RioExp := (d,h,n,k) -> coeftayl(d*h^k, x=0,n)*n!/k!:
%p A131222 A131222 := (n,k) -> RioExp(1,log((1-x)/(1-2*x)),n,k):
%p A131222 seq(print(seq(A131222(n,k),k=0..n)),n=0..5); # _Peter Luschny_, Apr 15 2015
%p A131222 # The function BellMatrix is defined in A264428.
%p A131222 BellMatrix(n -> `if`(n=0,1,n!*(2^(n+1)-1)), 9); # _Peter Luschny_, Jan 27 2016
%t A131222 BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A131222 rows = 12;
%t A131222 M = BellMatrix[If[# == 0, 1, #! (2^(#+1) - 1)]&, rows];
%t A131222 Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 24 2018, after _Peter Luschny_ *)
%o A131222 (Maxima) T(n,m):=if n=0 and m=0 then 1 else n!*sum((stirling1(k,m)*2^(n-k)*binomial(n-1,k-1))/k!,k,m,n); /* _Vladimir Kruchinin_, Sep 27 2012 */
%o A131222 (Sage)
%o A131222 def Lah(n, k):
%o A131222     if n == k: return 1
%o A131222     if k<0 or  k>n: return 0
%o A131222     return (k*n*gamma(n)^2)/(gamma(k+1)^2*gamma(n-k+1))
%o A131222 matrix(ZZ, 8, Lah) * matrix(ZZ, 8, stirling_number1) # as a square matrix _Peter Luschny_, Apr 12 2015
%o A131222 # alternatively:
%o A131222 (Sage) # uses[bell_matrix from A264428]
%o A131222 bell_matrix(lambda n: A029767(n+1), 10) # _Peter Luschny_, Jan 18 2016
%Y A131222 Cf. A000007, A002866, A029767, A079638, A079639, A079640.
%K A131222 easy,nonn,tabl
%O A131222 0,5
%A A131222 _Paul Barry_, Jun 18 2007