OFFSET
0,4
FORMULA
From Werner Schulte, Nov 26 2023: (Start)
The row polynomials p(n, x) = Sum_{k=0..n} T(n, k) * x^k satisfy:
a) p'(n, x) = n * p(n-1, x) where p' is the first derivative of p;
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.
T(n, 0) = A001333(n) for n >= 0 and T(n, k) = T(n-1, k-1) * n / k for 0 < k <= n.
G.f.: (1 - (1+x) * t) / (1 - (2+2*x) * t - (1-2*x-x^2) * t^2). (End)
EXAMPLE
Triangle T(n, k) starts:
[0] 1;
[1] 1, 1;
[2] 3, 2, 1;
[3] 7, 9, 3, 1;
[4] 17, 28, 18, 4, 1;
[5] 41, 85, 70, 30, 5, 1;
[6] 99, 246, 255, 140, 45, 6, 1;
[7] 239, 693, 861, 595, 245, 63, 7, 1;
[8] 577, 1912, 2772, 2296, 1190, 392, 84, 8, 1;
[9] 1393, 5193, 8604, 8316, 5166, 2142, 588, 108, 9, 1;
MAPLE
P := proc(n) option remember; ifelse(n <= 1, 1, 2*P(n - 1) + P(n - 2)) end:
T := (n, k) -> P(n - k) * binomial(n, k):
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;
MATHEMATICA
P[n_] := P[n] = If[n <= 1, 1, 2 P[n - 1] + P[n - 2]];
T[n_, k_] := P[n - k] Binomial[n, k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 10 2024, after Peter Luschny *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Nov 25 2023
STATUS
approved