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A130191 Square of the Stirling2 matrix A048993.

%I A130191 #61 Apr 19 2025 10:04:00
%S A130191 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 A130191 280,30,1,0,877,6230,8092,3465,595,42,1,0,4140,40819,70756,40992,
%U A130191 10010,1120,56,1,0,21147,283944,638423,479976,156072,24570,1932,72,1
%N A130191 Square of the Stirling2 matrix A048993.
%C A130191 Without row n=0 and column k=0 this is triangle A039810.
%C A130191 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 A130191 The triangle is also called the exponential Riordan array [1, exp(exp(x)-1)]. - _Peter Luschny_, Apr 19 2015
%C A130191 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 A130191 G. C. Greubel, <a href="/A130191/b130191.txt">Rows n=0..100 of triangle, flattened</a>
%H A130191 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 A130191 Wolfdieter Lang, <a href="/A130191/a130191.txt">First 10 rows and more</a>
%H A130191 John Riordan, <a href="/A002720/a002720_2.pdf">Letter, Apr 28 1976.</a> (See third page)
%F A130191 a(n,k) = Sum_{j=k..n} S2(n,j) * S2(j,k), n>=k>=0.
%F A130191 E.g.f. row polynomials with argument x: exp(x*f(f(z))).
%F A130191 E.g.f. column k: ((exp(exp(x) - 1) - 1)^k)/k!.
%e A130191 Triangle starts:
%e A130191   1;
%e A130191   0,   1;
%e A130191   0,   2,    1;
%e A130191   0,   5,    6,    1;
%e A130191   0,  15,   32,   12,    1;
%e A130191   0,  52,  175,  110,   20,   1;
%e A130191   0, 203, 1012,  945,  280,  30,  1;
%e A130191   0, 877, 6230, 8092, 3465, 595, 42, 1;
%p A130191 # The function BellMatrix is defined in A264428.
%p A130191 BellMatrix(n -> combinat:-bell(n+1), 9); # _Peter Luschny_, Jan 27 2016
%t A130191 BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A130191 rows = 10;
%t A130191 M = BellMatrix[BellB[# + 1]&, rows];
%t A130191 Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 23 2018, after _Peter Luschny_ *)
%t A130191 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 A130191 (Sage) # uses[riordan_array from A256893]
%o A130191 riordan_array(1, exp(exp(x) - 1), 8, exp=true) # _Peter Luschny_, Apr 19 2015
%o A130191 (PARI) for(n=0, 9, for(k=0, n, print1(sum(j=k, n, stirling(n, j, 2)*stirling(j, k, 2)), ", "))) \\ _G. C. Greubel_, Jul 10 2018
%Y A130191 Columns k=0..3 give A000007, A000110 (for n > 0), A000558, A000559.
%Y A130191 Row sums: A000258.
%Y A130191 Alternating row sums: A130410.
%Y A130191 T(2n,n) gives A321712.
%Y A130191 Cf. A039810 (another version), A048993.
%K A130191 nonn,tabl,easy
%O A130191 0,5
%A A130191 _Wolfdieter Lang_, Jun 01 2007