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A109822 Triangle read by rows: T(n,1)=1, T(n,k) = T(n-1,k) + (n-1)T(n-1, k-1) for 1 <= k <= n.

%I A109822 #15 Aug 17 2017 22:32:27
%S A109822 1,1,2,1,4,6,1,7,18,24,1,11,46,96,120,1,16,101,326,600,720,1,22,197,
%T A109822 932,2556,4320,5040,1,29,351,2311,9080,22212,35280,40320,1,37,583,
%U A109822 5119,27568,94852,212976,322560,362880,1,46,916,10366,73639,342964,1066644
%N A109822 Triangle read by rows: T(n,1)=1, T(n,k) = T(n-1,k) + (n-1)T(n-1, k-1) for 1 <= k <= n.
%C A109822 T(n,n) = n!. Sum of row n is the signless Stirling number of the first kind s(n,2)(A000254). T(n,k) = A096747(n,k) for 1 <= k <= n.
%F A109822 T(n, k) = Sum_{i=0..k-1} |stirling1(n, n-i)| for 1 <= k <= n.
%F A109822 From _Peter Bala_, Jul 08 2012: (Start)
%F A109822 E.g.f.: x/(1-x)*{1/(1-x*z)^(1/x) - 1/(1-x*z)} = x*z + (x + 2*x^2)*z^2/2! + (x + 4*x^2 + 6*x^3)*z^3/3! + ... Cf. the e.g.f. of A059518.
%F A109822 (End)
%e A109822 T(5,3) = 46 because 18 + 4*7 = 46.
%e A109822 Triangle begins:
%e A109822 Row 1:                    1
%e A109822 Row 2:                 1     2
%e A109822 Row 3:              1     4     6
%e A109822 Row 4:           1     7    18    24
%e A109822 Row 5:        1    11    46    96   120
%e A109822 Row 6:     1    16   101   326   600   720
%e A109822 Row 7:  1    22   197   932  2556  4320  5040
%p A109822 with(combinat): T:=(n,k)->add(abs(stirling1(n,n-i)),i=0..k-1): for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form T:=proc(n,k) if k=1 then 1 elif k=n then n! else T(n-1,k)+(n-1)*T(n-1,k-1) fi end: for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form. - _Emeric Deutsch_, Jul 03 2005
%p A109822 A109822_row := proc(n) local k,i;
%p A109822 add(add(abs(combinat[stirling1](n, n-i)), i=0..k)*x^(n-k-1),k=0..n-1);
%p A109822 seq(coeff(%,x,n-k),k=1..n) end:
%p A109822 seq(print(A109822_row(n)),n=1..7); # _Peter Luschny_, Sep 18 2011
%t A109822 Table[Sum[Abs@ StirlingS1[n, n - i], {i, 0, k - 1}], {n, 10}, {k, n}] // Flatten (* _Michael De Vlieger_, Aug 17 2017 *)
%Y A109822 Cf. A000254, A049444, A059518, A096747, A137650.
%K A109822 nonn,tabl
%O A109822 1,3
%A A109822 _Emeric Deutsch_, Jul 03 2005