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%I #43 Feb 13 2022 08:49:41
%S 1,-2,1,7,-6,1,-35,40,-12,1,228,-315,130,-20,1,-1834,2908,-1485,320,
%T -30,1,17582,-30989,18508,-5005,665,-42,1,-195866,375611,-253400,
%U 81088,-13650,1232,-56,1,2487832,-5112570,3805723,-1389612,279048,-32130,2100,-72,1
%N Matrix square of Stirling-1 triangle A008275.
%C Exponential Riordan array [1/((1 + x)*(1 + log(1 + x))), log(1 + log(1 + x))]. The row sums of the unsigned array give A007840 (apart from the initial term). - _Peter Bala_, Jul 22 2014
%C Also the Bell transform of (-1)^n*A003713(n+1). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 28 2016
%H Seiichi Manyama, <a href="/A039814/b039814.txt">Rows n = 1..140, flattened (Rows n = 1..60 from Vincenzo Librandi)</a>
%F E.g.f. of k-th column: ((log(1+log(1+x)))^k)/k!.
%F E.g.f.: 1/(1 + t)*( 1 + log(1 + t) )^(x-1) = 1 + (-2 + x)*t + (7 - 6*x + x^2)*t^2/2! + .... - _Peter Bala_, Jul 22 2014
%F T(n,k) = Sum_{j=0..n} Stirling1(n,j) * Stirling1(j,k). - _Seiichi Manyama_, Feb 13 2022
%e Triangle begins:
%e 1;
%e -2, 1;
%e 7, -6, 1;
%e -35, 40, -12, 1;
%e 228, -315, 130, -20, 1;
%e -1834, 2908, -1485, 320, -30, 1;
%e ...
%p # The function BellMatrix is defined in A264428.
%p # Adds (1,0,0,0, ..) as column 0.
%p BellMatrix(n -> (-1)^n*add(k!*abs(Stirling1(n+1,k+1)), k=0..n), 10); # _Peter Luschny_, Jan 28 2016
%t max = 9; t = Table[StirlingS1[n, k], {n, 1, max}, {k, 1, max}]; t2 = t.t; Table[t2[[n, k]], {n, 1, max}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Feb 01 2013 *)
%t rows = 9;
%t t = Table[(-1)^n*Sum[k!*Abs[StirlingS1[n+1, k+1]], {k,0,n}], {n, 0, rows}];
%t T[n_, k_] := BellY[n, k, t];
%t Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 22 2018, after _Peter Luschny_ *)
%o (PARI) T(n, k) = sum(j=0, n, stirling(n, j, 1)*stirling(j, k, 1)); \\ _Seiichi Manyama_, Feb 13 2022
%Y Column k=1..3 give (-1)^(n-1) * A003713(n), (-1)^n * A341587(n), (-1)^(n-1) * A341588(n).
%Y Cf. A007840.
%Y Cf. A039810, A039815, A039816, A039817.
%K sign,tabl,nice
%O 1,2
%A _Christian G. Bower_, Feb 15 1999
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