A371740 - OEIS

%I #9 Apr 14 2024 13:48:03

%S 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 10,0,54,105,65,15,1,0,288,600,420,120,12,0,264,574,435,145,21,1,0,

%U 1680,3836,3150,1190,210,14,0,1560,3682,3199,1330,280,28,1,0,11520,28224,25984,11760,2800,336,16

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

%F 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 Row polynomials: R(2*n, x) =  (2*n - 1 + x) * Product_{i = 0..n-2} (x + i) for n >= 1.

%F R(2*n+1, x) = (2*n + 2) * Product_{i = 0..n-1} (x + i) for n >= 0.

%F T(2*n, k) = |Stirling1(n, k)| + n*|Stirling1(n-1, k)| = A132393(n, k) + n* A132393(n-1, k);

%F T(2*n+1, k) = (2*n + 2)*|Stirling1(n, k)| = (2*n + 2)*A132393(n, k).

%F n-th row sum equals 2 * floor((n+1)/2)! for n >= 1.

%e Triangle begins

%e  n\k |  0    1     2    3    4   5

%e  - - - - - - - - - - - - - - - - -

%e   0  |  1

%e   1  |  2

%e   2  |  1    1

%e   3  |  0    4

%e   4  |  0    3     1

%e   5  |  0    6     6

%e   6  |  0    5     6    1

%e   7  |  0   16    24    8

%e   8  |  0   14    23   10    1

%e   9  |  0   60   110   60   10

%e  10  |  0   54   105   65   15   1

%e  ...

%p with(combinat):

%p 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 seq(print(seq(T(n, k), k = 0..floor(n/2))), n = 0..12);

%Y Cf. A052849, A132393, A371741.

%K nonn,tabf,easy

%O 0,2

%A _Peter Bala_, Apr 09 2024