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%I #40 Apr 17 2021 03:40:51
%S 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,
%T 280,30,1,0,877,6230,8092,3465,595,42,1,0,4140,40819,70756,40992,
%U 10010,1120,56,1,0,21147,283944,638423,479976,156072,24570,1932,72,1
%N Square of the Stirling2 matrix A048993.
%C Without row n=0 and column m=0 this is triangle A039810.
%C 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.
%C The triangle is also called the exponential Riordan array [1, exp(exp(x)-1)]. - _Peter Luschny_, Apr 19 2015
%C 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
%H G. C. Greubel, <a href="/A130191/b130191.txt">Rows n=0..100 of triangle, flattened</a>
%H Marin Knežević, Vedran Krčadinac, and Lucija Relić, <a href="https://arxiv.org/abs/2012.15307">Matrix products of binomial coefficients and unsigned Stirling numbers</a>, arXiv:2012.15307 [math.CO], 2020.
%H Wolfdieter Lang, <a href="/A130191/a130191.txt">First 10 rows and more</a>
%H John Riordan, <a href="/A002720/a002720_2.pdf">Letter, Apr 28 1976.</a> (See third page)
%F a(n,m) = Sum_{k=m..n} S2(n,k) * S2(k,m), n>=m>=0.
%F E.g.f. row polynomials with argument x: exp(x*f(f(z))).
%F E.g.f. column m: ((exp(exp(x)-1)-1)^m)/m!.
%e Triangle starts:
%e [1]
%e [0, 1]
%e [0, 2, 1]
%e [0, 5, 6, 1]
%e [0,15,32,12,1]
%p # The function BellMatrix is defined in A264428.
%p BellMatrix(n -> combinat:-bell(n+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 = 10;
%t M = BellMatrix[BellB[# + 1]&, rows];
%t Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 23 2018, after _Peter Luschny_ *)
%t 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 *)
%o (Sage) # uses[riordan_array from A256893]
%o riordan_array(1, exp(exp(x) - 1), 8, exp=true) # _Peter Luschny_, Apr 19 2015
%o (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
%Y Row sums: A000258. Alternating row sums: A130410.
%Y Cf. A000110, A039810, A048993.
%Y T(2n,n) gives A321712.
%K nonn,tabl,easy
%O 0,5
%A _Wolfdieter Lang_, Jun 01 2007
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