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A367564 Triangular array read by rows: T(n, k) = binomial(n, k) * A001333(n - k).

%I A367564 #13 Oct 10 2024 08:18:24
%S A367564 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 A367564 1,239,693,861,595,245,63,7,1,577,1912,2772,2296,1190,392,84,8,1,1393,
%U A367564 5193,8604,8316,5166,2142,588,108,9,1,3363,13930,25965,28680,20790,10332,3570,840,135,10,1
%N A367564 Triangular array read by rows: T(n, k) = binomial(n, k) * A001333(n - k).
%F A367564 From _Werner Schulte_, Nov 26 2023: (Start)
%F A367564 The row polynomials p(n, x) = Sum_{k=0..n} T(n, k) * x^k satisfy:
%F A367564   a) p'(n, x) = n * p(n-1, x) where p' is the first derivative of p;
%F A367564   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 A367564 T(n, 0) = A001333(n) for n >= 0 and T(n, k) = T(n-1, k-1) * n / k for 0 < k <= n.
%F A367564 G.f.: (1 - (1+x) * t) / (1 - (2+2*x) * t - (1-2*x-x^2) * t^2). (End)
%e A367564 Triangle T(n, k) starts:
%e A367564 [0]    1;
%e A367564 [1]    1,    1;
%e A367564 [2]    3,    2,    1;
%e A367564 [3]    7,    9,    3,    1;
%e A367564 [4]   17,   28,   18,    4,    1;
%e A367564 [5]   41,   85,   70,   30,    5,    1;
%e A367564 [6]   99,  246,  255,  140,   45,    6,   1;
%e A367564 [7]  239,  693,  861,  595,  245,   63,   7,   1;
%e A367564 [8]  577, 1912, 2772, 2296, 1190,  392,  84,   8, 1;
%e A367564 [9] 1393, 5193, 8604, 8316, 5166, 2142, 588, 108, 9, 1;
%p A367564 P := proc(n) option remember; ifelse(n <= 1, 1, 2*P(n - 1) + P(n - 2)) end:
%p A367564 T := (n, k) -> P(n - k) * binomial(n, k):
%p A367564 for n from 0 to 9 do seq(T(n, k), k = 0..n) od;
%t A367564 P[n_] := P[n] = If[n <= 1, 1, 2 P[n - 1] + P[n - 2]];
%t A367564 T[n_, k_] := P[n - k] Binomial[n, k];
%t A367564 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Oct 10 2024, after _Peter Luschny_ *)
%Y A367564 Cf. A001333 (column 0), A006012 (row sums), A367211.
%K A367564 nonn,tabl
%O A367564 0,4
%A A367564 _Peter Luschny_, Nov 25 2023