|
%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
|
