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Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.
1

%I #21 Feb 12 2022 14:16:20

%S 1,3,1,14,9,1,90,83,18,1,744,870,275,30,1,7560,10474,4275,685,45,1,

%T 91440,143892,70924,14805,1435,63,1,1285200,2233356,1274196,324289,

%U 41160,2674,84,1,20603520,38769840,24870572,7398972,1151409,98280,4578

%N Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.

%C Matrix product of unsigned Lah-triangle |A008297(n,k)| and Stirling1-triangle A008275(n,k) is unsigned Stirling1-triangle |A008275(n,k)|.

%C Also the Bell transform of A029767(n+1). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 26 2016

%C Essentially the same as A131222. - _Peter Bala_, Feb 12 2022

%F T(n, k) = Sum_{i=k..n} |A008297(n, i)| * |A008275(i, k)|.

%F E.g.f.: ((1-x)/(1-2*x))^y. - _Vladeta Jovovic_, Nov 22 2003

%e Triangle begins

%e 1;

%e 3, 1;

%e 14, 9, 1;

%e 90, 83, 18, 1;

%e 744, 870, 275, 30, 1;

%e 7560, 10474, 4275, 685, 45, 1;

%e ...

%p # The function BellMatrix is defined in A264428.

%p # Adds (1, 0, 0, 0, ..) as column 0.

%p BellMatrix(n -> n!*(2^(n+1)-1), 9); # _Peter Luschny_, Jan 26 2016

%t BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];

%t B = BellMatrix[Function[n, n! (2^(n + 1) - 1)], rows = 12];

%t Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)

%Y Cf. A002866 (row sums), A029767 (first column), A131222.

%K nonn,tabl

%O 1,2

%A _Vladeta Jovovic_, Jan 30 2003