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A079638 Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.

%I A079638 #27 Aug 07 2025 12:36:56
%S A079638 1,3,1,14,9,1,90,83,18,1,744,870,275,30,1,7560,10474,4275,685,45,1,
%T A079638 91440,143892,70924,14805,1435,63,1,1285200,2233356,1274196,324289,
%U A079638 41160,2674,84,1,20603520,38769840,24870572,7398972,1151409,98280,4578
%N A079638 Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.
%C A079638 Matrix product of unsigned Lah-triangle |A008297(n,k)| and Stirling1-triangle A008275(n,k) is unsigned Stirling1-triangle |A008275(n,k)|.
%C A079638 Also the Bell transform of A029767(n+1). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 26 2016
%C A079638 Essentially the same as A131222. - _Peter Bala_, Feb 12 2022
%H A079638 Michael De Vlieger, <a href="/A079638/b079638.txt">Table of n, a(n) for n = 1..1225</a> (rows n = 1..50, flattened).
%H A079638 William Keith, Rishi Nath, and James Sellers, <a href="https://arxiv.org/abs/2508.00074">On simultaneous (s, s+t, s+2t, ...)-core partitions</a>, arXiv:2508.00074 [math.CO], 2025. See p. 3.
%F A079638 T(n, k) = Sum_{i=k..n} |A008297(n, i)| * |A008275(i, k)|.
%F A079638 E.g.f.: ((1-x)/(1-2*x))^y. - _Vladeta Jovovic_, Nov 22 2003
%e A079638 Triangle begins
%e A079638      1;
%e A079638      3,     1;
%e A079638     14,     9,    1;
%e A079638     90,    83,   18,   1;
%e A079638    744,   870,  275,  30,  1;
%e A079638   7560, 10474, 4275, 685, 45, 1;
%e A079638   ...
%p A079638 # The function BellMatrix is defined in A264428.
%p A079638 # Adds (1, 0, 0, 0, ..) as column 0.
%p A079638 BellMatrix(n -> n!*(2^(n+1)-1), 9); # _Peter Luschny_, Jan 26 2016
%t A079638 BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A079638 B = BellMatrix[Function[n, n! (2^(n + 1) - 1)], rows = 12];
%t A079638 Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)
%Y A079638 Cf. A002866 (row sums), A029767 (first column), A131222.
%K A079638 nonn,tabl
%O A079638 1,2
%A A079638 _Vladeta Jovovic_, Jan 30 2003