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%I #9 Nov 26 2023 15:54:47
%S 1,1,1,3,2,1,7,9,3,1,17,28,18,4,1,41,85,70,30,5,1,99,246,255,140,45,6,
%T 1,239,693,861,595,245,63,7,1,577,1912,2772,2296,1190,392,84,8,1,1393,
%U 5193,8604,8316,5166,2142,588,108,9,1,3363,13930,25965,28680,20790,10332,3570,840,135,10,1
%N Triangular array read by rows: T(n, k) = binomial(n, k) * A001333(n - k).
%F From _Werner Schulte_, Nov 26 2023: (Start)
%F The row polynomials p(n, x) = Sum_{k=0..n} T(n, k) * x^k satisfy:
%F a) p'(n, x) = n * p(n-1, x) where p' is the first derivative of p;
%F b) p(0, x) = 1, p(1, x) = 1 + x and p(n, x) = (2+2*x) * p(n-1, x) + (1-2*x-x^2) * p(n-2, x) for n > 1.
%F T(n, 0) = A001333(n) for n >= 0 and T(n, k) = T(n-1, k-1) * n / k for 0 < k <= n.
%F G.f.: (1 - (1+x) * t) / (1 - (2+2*x) * t - (1-2*x-x^2) * t^2). (End)
%e Triangle T(n, k) starts:
%e [0] 1;
%e [1] 1, 1;
%e [2] 3, 2, 1;
%e [3] 7, 9, 3, 1;
%e [4] 17, 28, 18, 4, 1;
%e [5] 41, 85, 70, 30, 5, 1;
%e [6] 99, 246, 255, 140, 45, 6, 1;
%e [7] 239, 693, 861, 595, 245, 63, 7, 1;
%e [8] 577, 1912, 2772, 2296, 1190, 392, 84, 8, 1;
%e [9] 1393, 5193, 8604, 8316, 5166, 2142, 588, 108, 9, 1;
%p P := proc(n) option remember; ifelse(n <= 1, 1, 2*P(n - 1) + P(n - 2)) end:
%p T := (n, k) -> P(n - k) * binomial(n, k):
%p for n from 0 to 9 do seq(T(n, k), k = 0..n) od;
%Y Cf. A001333 (column 0), A006012 (row sums), A367211.
%K nonn,tabl
%O 0,4
%A _Peter Luschny_, Nov 25 2023
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