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

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, 1, 239, 693, 861, 595, 245, 63, 7, 1, 577, 1912, 2772, 2296, 1190, 392, 84, 8, 1, 1393, 5193, 8604, 8316, 5166, 2142, 588, 108, 9, 1, 3363, 13930, 25965, 28680, 20790, 10332, 3570, 840, 135, 10, 1

Offset

0,4

Links

Table of n, a(n) for n=0..65.

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

Cf. A001333 (column 0), A006012 (row sums), A367211.

Sequence in context: A145035 A359413 A192020 * A171128 A122832 A056151

Adjacent sequences: A367561 A367562 A367563 * A367565 A367566 A367567

Keyword

nonn,tabl

Author

Peter Luschny, Nov 25 2023

Status

approved