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A059374 Triangle read by rows, T(n, k) = Sum_{i=0..n} L'(n, n-i) * binomial(i, k), for k = 0..n-1.

%I A059374 #18 Mar 28 2023 07:42:55
%S A059374 1,3,2,13,18,6,73,156,108,24,501,1460,1560,720,120,4051,15030,21900,
%T A059374 15600,5400,720,37633,170142,315630,306600,163800,45360,5040,394353,
%U A059374 2107448,4763976,5891760,4292400,1834560,423360,40320
%N A059374 Triangle read by rows, T(n, k) = Sum_{i=0..n} L'(n, n-i) * binomial(i, k), for k = 0..n-1.
%C A059374 L'(n, i) are unsigned Lah numbers (Cf. A008297).
%H A059374 G. C. Greubel, <a href="/A059374/b059374.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A059374 E.g.f.: exp(x/(1-(1+y)*x))/(1-(1+y)*x)^2. - _Vladeta Jovovic_, May 10 2003
%e A059374 Triangle begins:
%e A059374   [1],
%e A059374   [3, 2],
%e A059374   [13, 18, 6],
%e A059374   [73, 156, 108, 24],
%e A059374   [501, 1460, 1560, 720, 120],
%e A059374   ...
%t A059374 t[n_, k_] := Sum[ Binomial[n-1, n-i-1]*n!/(n-i)!*Binomial[i, k], {i, 0, n}]; Table[t[n, k], {n, 1, 8}, {k, 0, n-1}] // Flatten (* _Jean-François Alcover_, Mar 22 2013 *)
%o A059374 (PARI) for(n=1,10, for(k=0,n-1, print1(sum(j=0,n, binomial(j,k)* binomial(n-1,n-j-1)*n!/(n-j)!), ", "))) \\ _G. C. Greubel_, Jan 29 2018
%Y A059374 Cf. T(n, 0) = A000262, A025168 (row sums), A000012 (alternating row sums), A059110.
%Y A059374 Cf. A008297, A271703.
%K A059374 easy,nonn,tabl
%O A059374 1,2
%A A059374 _Vladeta Jovovic_, Jan 28 2001