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A131222
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Exponential Riordan array [1, log((1-x)/(1-2x))].
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5
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1, 0, 1, 0, 3, 1, 0, 14, 9, 1, 0, 90, 83, 18, 1, 0, 744, 870, 275, 30, 1, 0, 7560, 10474, 4275, 685, 45, 1, 0, 91440, 143892, 70924, 14805, 1435, 63, 1, 0, 1285200, 2233356, 1274196, 324289, 41160, 2674, 84, 1
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OFFSET
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0,5
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COMMENTS
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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
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LINKS
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FORMULA
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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
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EXAMPLE
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Number triangle starts:
1,
0, 1;
0, 3, 1;
0, 14, 9, 1;
0, 90, 83, 18, 1;
0, 744, 870, 275, 30, 1;
...
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MAPLE
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RioExp := (d, h, n, k) -> coeftayl(d*h^k, x=0, n)*n!/k!:
A131222 := (n, k) -> RioExp(1, log((1-x)/(1-2*x)), n, k):
# The function BellMatrix is defined in A264428.
BellMatrix(n -> `if`(n=0, 1, n!*(2^(n+1)-1)), 9); # Peter Luschny, Jan 27 2016
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MATHEMATICA
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BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[If[# == 0, 1, #! (2^(#+1) - 1)]&, rows];
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PROG
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(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 */
(Sage)
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, Lah) * matrix(ZZ, 8, stirling_number1) # as a square matrix Peter Luschny, Apr 12 2015
# alternatively:
(Sage) # uses[bell_matrix from A264428]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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