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A130191
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Square of the Stirling2 matrix A048993.
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10
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1, 0, 1, 0, 2, 1, 0, 5, 6, 1, 0, 15, 32, 12, 1, 0, 52, 175, 110, 20, 1, 0, 203, 1012, 945, 280, 30, 1, 0, 877, 6230, 8092, 3465, 595, 42, 1, 0, 4140, 40819, 70756, 40992, 10010, 1120, 56, 1, 0, 21147, 283944, 638423, 479976, 156072, 24570, 1932, 72, 1
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OFFSET
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0,5
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COMMENTS
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Without row n=0 and column m=0 this is triangle A039810.
This is an associated Sheffer matrix with e.g.f. of the m-th column (exp(f(x)-1))^m)/m! with f(x)=:exp(x)-1.
The triangle is also called the exponential Riordan array [1, exp(exp(x)-1)]. - Peter Luschny, Apr 19 2015
Also the Bell transform of shifted Bell numbers A000110(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016
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LINKS
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FORMULA
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a(n,m) = Sum_{k=m..n} S2(n,k) * S2(k,m), n>=m>=0.
E.g.f. row polynomials with argument x: exp(x*f(f(z))).
E.g.f. column m: ((exp(exp(x)-1)-1)^m)/m!.
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EXAMPLE
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Triangle starts:
[1]
[0, 1]
[0, 2, 1]
[0, 5, 6, 1]
[0,15,32,12,1]
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MAPLE
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# The function BellMatrix is defined in A264428.
BellMatrix(n -> combinat:-bell(n+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 = 10;
M = BellMatrix[BellB[# + 1]&, rows];
a[n_, m_]:= Sum[StirlingS2[n, k]*StirlingS2[k, m], {k, m, n}]; Table[a[n, m], {n, 0, 100}, {m, 0, n}]//Flatten (* G. C. Greubel, Jul 10 2018 *)
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PROG
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(Sage) # uses[riordan_array from A256893]
riordan_array(1, exp(exp(x) - 1), 8, exp=true) # Peter Luschny, Apr 19 2015
(PARI) for(n=0, 10, for(m=0, n, print1(sum(k=m, n, stirling(n, k, 2)* stirling(k, m, 2)), ", "))) \\ G. C. Greubel, Jul 10 2018
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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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