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Exponential Riordan array [1, log((1-x)/(1-2x))].
5

%I #43 Jan 09 2024 12:18:46

%S 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 4275,685,45,1,0,91440,143892,70924,14805,1435,63,1,0,1285200,2233356,

%U 1274196,324289,41160,2674,84,1

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

%C 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 The Bell transform of A029767(n+1). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 18 2016

%C Essentially the same as A079638. - _Peter Bala_, Feb 12 2022

%H Alois P. Heinz, <a href="/A131222/b131222.txt">Rows n = 0..140, flattened</a>

%H 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 Row sums are A002866.

%F Second column is A029767.

%F 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 Number triangle starts:

%e 1,

%e 0, 1;

%e 0, 3, 1;

%e 0, 14, 9, 1;

%e 0, 90, 83, 18, 1;

%e 0, 744, 870, 275, 30, 1;

%e ...

%p RioExp := (d,h,n,k) -> coeftayl(d*h^k, x=0,n)*n!/k!:

%p A131222 := (n,k) -> RioExp(1,log((1-x)/(1-2*x)),n,k):

%p seq(print(seq(A131222(n,k),k=0..n)),n=0..5); # _Peter Luschny_, Apr 15 2015

%p # The function BellMatrix is defined in A264428.

%p BellMatrix(n -> `if`(n=0,1,n!*(2^(n+1)-1)), 9); # _Peter Luschny_, Jan 27 2016

%t BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];

%t rows = 12;

%t M = BellMatrix[If[# == 0, 1, #! (2^(#+1) - 1)]&, rows];

%t Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 24 2018, after _Peter Luschny_ *)

%o (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 (Sage)

%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, Lah) * matrix(ZZ, 8, stirling_number1) # as a square matrix _Peter Luschny_, Apr 12 2015

%o # alternatively:

%o (Sage) # uses[bell_matrix from A264428]

%o bell_matrix(lambda n: A029767(n+1), 10) # _Peter Luschny_, Jan 18 2016

%Y Cf. A000007, A002866, A029767, A079638, A079639, A079640.

%K easy,nonn,tabl

%O 0,5

%A _Paul Barry_, Jun 18 2007