A371741 Triangle read by rows: g.f. (1 - t)^(-x) * (1 + t)^(3-x).
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, 0, 114, 215, 125, 25, 1, 0, 102, 205, 135, 35, 3, 0, 552, 1174, 855, 265, 33, 1, 0, 504, 1122, 885, 315, 51, 3, 0, 3240, 7518, 6349, 2520, 490, 42, 1, 0, 3000, 7210, 6447, 2800, 630, 70, 3
Offset: 0
Examples
Triangle begins n\k | 0 1 2 3 4 5 - - - - - - - - - - - - - - - - - - - - 0 | 1 1 | 3 2 | 3 1 3 | 1 3 4 | 0 7 1 5 | 0 5 3 6 | 0 11 12 1 7 | 0 9 12 3 8 | 0 30 47 18 1 9 | 0 26 45 22 3 10 | 0 114 215 125 25 1 11 | 0 102 205 135 35 3 ...
Programs
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Maple
with(combinat): 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: seq(print(seq(T(n, k), k = 0..floor(n/2))), n = 0..12);
Formula
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! + ....
Row polynomials: R(2*n, x) = (4*n - 1 + x) * Product_{i = 0..n-2} (x + i) for n >= 1.
R(2*n+1, x) = (4*n - 3 + 3*x) * Product_{i = 0..n-2} (x + i) for n >= 1.
n-th row sums equals 4*floor(n/2)! for n >= 2.