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A371741 Triangle read by rows: g.f. (1 - t)^(-x) * (1 + t)^(3-x).

%I A371741 #10 Apr 14 2024 13:47:48
%S A371741 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 A371741 0,114,215,125,25,1,0,102,205,135,35,3,0,552,1174,855,265,33,1,0,504,
%U A371741 1122,885,315,51,3,0,3240,7518,6349,2520,490,42,1,0,3000,7210,6447,2800,630,70,3
%N A371741 Triangle read by rows: g.f. (1 - t)^(-x) * (1 + t)^(3-x).
%F A371741 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 A371741 Row polynomials: R(2*n, x) =  (4*n - 1 + x) * Product_{i = 0..n-2} (x + i) for n >= 1.
%F A371741 R(2*n+1, x) = (4*n - 3 + 3*x) * Product_{i = 0..n-2} (x + i) for n >= 1.
%F A371741 T(2*n, k) = |Stirling1(n, k)| + 3*n*|Stirling1(n-1, k)| = A132393(n, k) + 3*n*A132393(n-1, k).
%F A371741 T(2*n+1, k) = 3*|Stirling1(n, k)| + n*|Stirling1(n-1, k)| = 3*A132393(n, k) + n*A132393(n-1, k).
%F A371741 T(2*n, k) = (4*n - 1)*A132393(n-1, k) + A132393(n-1, k-1).
%F A371741 T(2*n+1, k) = (4*n - 3)*A132393(n-1, k) + 3*A132393(n-1, k-1).
%F A371741 n-th row sums equals 4*floor(n/2)! for n >= 2.
%e A371741 Triangle begins
%e A371741   n\k |  0     1     2     3     4      5
%e A371741   - - - - - - - - - - - - - - - - - - - -
%e A371741    0  |  1
%e A371741    1  |  3
%e A371741    2  |  3     1
%e A371741    3  |  1     3
%e A371741    4  |  0     7     1
%e A371741    5  |  0     5     3
%e A371741    6  |  0    11    12     1
%e A371741    7  |  0     9    12     3
%e A371741    8  |  0    30    47    18     1
%e A371741    9  |  0    26    45    22     3
%e A371741   10  |  0   114   215   125    25     1
%e A371741   11  |  0   102   205   135    35     3
%e A371741   ...
%p A371741 with(combinat):
%p A371741 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 A371741 seq(print(seq(T(n, k), k = 0..floor(n/2))), n = 0..12);
%Y A371741 Cf. A130534, A132393, A371740.
%K A371741 nonn,tabf,easy
%O A371741 0,2
%A A371741 _Peter Bala_, Apr 05 2024