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A371740 Triangle read by rows: g.f. (1 - t)^(-x) * (1 + t)^(2-x).

%I A371740 #9 Apr 14 2024 13:48:03
%S A371740 1,2,1,1,0,4,0,3,1,0,6,6,0,5,6,1,0,16,24,8,0,14,23,10,1,0,60,110,60,
%T A371740 10,0,54,105,65,15,1,0,288,600,420,120,12,0,264,574,435,145,21,1,0,
%U A371740 1680,3836,3150,1190,210,14,0,1560,3682,3199,1330,280,28,1,0,11520,28224,25984,11760,2800,336,16
%N A371740 Triangle read by rows: g.f. (1 - t)^(-x) * (1 + t)^(2-x).
%F A371740 G.f.: (1 - t)^(-x)*(1 + t)^(2-x) = Sum_{n >= 0} R(n, x)*t^n/floor((n+1)/2)! = 1 + 2*t/1! + (1 + x)*t^2/1! + 4*x*t^3/2! + x*(3 + x)*t^4/2! + 6*x*(1 + x)*t^5/3! + x*(1 + x)*(5 + x)*t^6/3! + 8*x*(1 + x)*(2 + x)*t^7/3! + x*(1 + x)*(2 + x)*(7 + x)*t^8/4! + 10*x*(1 + x)*(2 + x)*(3 + x)*t^9/5! + ....
%F A371740 Row polynomials: R(2*n, x) =  (2*n - 1 + x) * Product_{i = 0..n-2} (x + i) for n >= 1.
%F A371740 R(2*n+1, x) = (2*n + 2) * Product_{i = 0..n-1} (x + i) for n >= 0.
%F A371740 T(2*n, k) = |Stirling1(n, k)| + n*|Stirling1(n-1, k)| = A132393(n, k) + n* A132393(n-1, k);
%F A371740 T(2*n+1, k) = (2*n + 2)*|Stirling1(n, k)| = (2*n + 2)*A132393(n, k).
%F A371740 n-th row sum equals 2 * floor((n+1)/2)! for n >= 1.
%e A371740 Triangle begins
%e A371740  n\k |  0    1     2    3    4   5
%e A371740  - - - - - - - - - - - - - - - - -
%e A371740   0  |  1
%e A371740   1  |  2
%e A371740   2  |  1    1
%e A371740   3  |  0    4
%e A371740   4  |  0    3     1
%e A371740   5  |  0    6     6
%e A371740   6  |  0    5     6    1
%e A371740   7  |  0   16    24    8
%e A371740   8  |  0   14    23   10    1
%e A371740   9  |  0   60   110   60   10
%e A371740  10  |  0   54   105   65   15   1
%e A371740  ...
%p A371740 with(combinat):
%p A371740 T := proc (n, k); if irem(n, 2) = 0 then abs(Stirling1((1/2)*n, k)) + (n/2)*abs(Stirling1((n-2)/2, k)) else (n+1)*abs(Stirling1((n-1)/2, k)) end if; end proc:
%p A371740 seq(print(seq(T(n, k), k = 0..floor(n/2))), n = 0..12);
%Y A371740 Cf. A052849, A132393, A371741.
%K A371740 nonn,tabf,easy
%O A371740 0,2
%A A371740 _Peter Bala_, Apr 09 2024