A371741 - OEIS

%I #10 Apr 14 2024 13:47:48

%S 1,3,3,1,1,3,0,7,1,0,5,3,0,11,12,1,0,9,12,3,0,30,47,18,1,0,26,45,22,3,

%T 0,114,215,125,25,1,0,102,205,135,35,3,0,552,1174,855,265,33,1,0,504,

%U 1122,885,315,51,3,0,3240,7518,6349,2520,490,42,1,0,3000,7210,6447,2800,630,70,3

%N Triangle read by rows: g.f. (1 - t)^(-x) * (1 + t)^(3-x).

%F G.f.: (1 - t)^(-x)*(1 + t)^(3-x) = Sum_{n >= 0} R(n, x)*t^n/floor(n/2)! = 1 + 3*t + (3 + x)^t^2/1! + (1 + 3*x)*t^3/1! + x*(7 + x)*t^4/2! + x*(5 + 3*x)*t^5/2! + x*(1 + x)*(11 + x)*t^6/3! + x*(1 + x)*(9 + 3*x)*t^7/3! + x*(1 + x)*(2 + x)*(15 + x)*t^8/4! + x*(1 + x)*(2 + x)*(13 + 3*x)*t^9/4! + ....

%F Row polynomials: R(2*n, x) =  (4*n - 1 + x) * Product_{i = 0..n-2} (x + i) for n >= 1.

%F R(2*n+1, x) = (4*n - 3 + 3*x) * Product_{i = 0..n-2} (x + i) for n >= 1.

%F T(2*n, k) = |Stirling1(n, k)| + 3*n*|Stirling1(n-1, k)| = A132393(n, k) + 3*n*A132393(n-1, k).

%F T(2*n+1, k) = 3*|Stirling1(n, k)| + n*|Stirling1(n-1, k)| = 3*A132393(n, k) + n*A132393(n-1, k).

%F T(2*n, k) = (4*n - 1)*A132393(n-1, k) + A132393(n-1, k-1).

%F T(2*n+1, k) = (4*n - 3)*A132393(n-1, k) + 3*A132393(n-1, k-1).

%F n-th row sums equals 4*floor(n/2)! for n >= 2.

%e Triangle begins

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

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

%e    0  |  1

%e    1  |  3

%e    2  |  3     1

%e    3  |  1     3

%e    4  |  0     7     1

%e    5  |  0     5     3

%e    6  |  0    11    12     1

%e    7  |  0     9    12     3

%e    8  |  0    30    47    18     1

%e    9  |  0    26    45    22     3

%e   10  |  0   114   215   125    25     1

%e   11  |  0   102   205   135    35     3

%e   ...

%p with(combinat):

%p T := proc (n, k); if irem(n, 2) = 0 then abs(Stirling1((1/2)*n, k)) + (3*n/2)*abs(Stirling1((n-2)/2, k)) else 3*abs(Stirling1((n-1)/2, k)) + ((n-1)/2)*abs(Stirling1((n-3)/2, k)) end if; end proc:

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

%Y Cf. A130534, A132393, A371740.

%K nonn,tabf,easy

%O 0,2

%A _Peter Bala_, Apr 05 2024