A367211 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 2 + 2x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 2x - x^2.

1, 2, 2, 5, 6, 3, 12, 20, 12, 4, 29, 60, 50, 20, 5, 70, 174, 180, 100, 30, 6, 169, 490, 609, 420, 175, 42, 7, 408, 1352, 1960, 1624, 840, 280, 56, 8, 985, 3672, 6084, 5880, 3654, 1512, 420, 72, 9, 2378, 9850, 18360, 20280, 14700, 7308, 2520, 600, 90, 10

Offset

1,2

Comments

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

Links

Table of n, a(n) for n=1..55.

Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.

Formula

p(n, x) = u*p(n-1, x) + v*p(n-2, x) for n >= 3, where p(1, x) = 1, p(2, x) = 2 + 2 x, u = p(2, x), and v = 1 - 2x - x^2.

p(n, x) = k*(b^n - c^n), where k = sqrt(1/8), b = x + 1 - sqrt(2), c = x + 1 + sqrt(2).

From Werner Schulte, Nov 24 2023 and Nov 25 2023: (Start)

The row polynomials p(n, x) = Sum_{k=0..n-1} T(n, k) * x^k satisfy the equation p'(n, x) = n * p(n-1, x) where p' is the first derivative of p.

T(n, k) = T(n-1, k-1) * n / k for 0 < k < n and T(n, 0) = A000129(n) for n > 0.

T(n, k) = A000129(n-k) * binomial(n, k) for 0 <= k < n.

G.f.: t / (1 - (2+2*x) * t - (1-2*x-x^2) * t^2). (End)

Example

First nine rows:
  [n\k] 0     1     2     3     4     5    6   7  8
  [1]   1;
  [2]   2     2;
  [3]   5     6    3;
  [4]  12    20    12     4;
  [5]  29    60    50    20     5;
  [6]  70   174   180   100    30     6;
  [7] 169   490   609   420   175    42   7;
  [8] 408  1352  1960  1624   840   280   56   8;
  [9] 985  3672  6084  5880  3654  1512  420  72  9;
.
Row 4 represents the polynomial p(4,x) = 12 + 20 x + 12 x^2 + 4 x^3, so that (T(4,k)) = (12, 20, 12, 4), k = 0..3.

Maple

P := proc(n) option remember; ifelse(n <= 1, n, 2*P(n - 1) + P(n - 2)) end:
T := (n, k) -> P(n - k) * binomial(n, k):
for n from 1 to 9 do [n], seq(T(n, k), k = 0..n-1) od;
# (after Werner Schulte)  Peter Luschny, Nov 24 2023

Mathematica

p[1, x_] := 1; p[2, x_] := 2 + 2 x; u[x_] := p[2, x]; v[x_] := 1 - 2 x - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

Crossrefs

Cf. A000129 (column 1), A361732 (column 2), A000027 (T(n,n-1)), A007070 (row sums, p(n,1)), A077957 (alternating row sums, p(n,-1)), A081179 (p(n,2), A077985 (p(n,-2), A081180 (p(n,3)), A007070 (p(n,-3)), A081182 (p(n,4)), A094440, A367208, A367209, A367210.

Sequence in context: A000403 A068763 A176989 * A250303 A368554 A301477

Adjacent sequences: A367208 A367209 A367210 * A367212 A367213 A367214

Keyword

nonn,tabl

Author

Clark Kimberling, Nov 13 2023

Status

approved