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A256893
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Exponential Riordan array [1, 1/(2-e^x)-1].
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68
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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
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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 Stirling set numbers and the unsigned Lah numbers.
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LINKS
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FORMULA
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T(n,k) = n!/k! * [x^n] (1/(2-exp(x))-1)^k. - Alois P. Heinz, Apr 17 2015
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EXAMPLE
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Number triangle starts:
1;
0, 1;
0, 3, 1;
0, 13, 9, 1;
0, 75, 79, 18, 1;
0, 541, 765, 265, 30, 1;
...
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MAPLE
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T:= (n, k)-> n!*coeff(series((1/(2-exp(x))-1)^k/k!, x, n+1), x, n):
# The function BellMatrix is defined in A264428.
BellMatrix(n -> polylog(-n-1, 1/2)/2, 9); # Peter Luschny, Jan 29 2016
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MATHEMATICA
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T[n_, k_] := n!*SeriesCoefficient[(1/(2 - Exp[x]) - 1)^k/k!, {x, 0, n}];
(* The function BellMatrix is defined in A264428. *)
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[#])-1&, 10, True] // Flatten (* Jean-François Alcover, Jul 16 2019, after Sage program *)
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PROG
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(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)) - 1, 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)
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CROSSREFS
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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.
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KEYWORD
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AUTHOR
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STATUS
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approved
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