This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A367211 #35 Mar 28 2025 04:12:08
%S A367211 1,2,2,5,6,3,12,20,12,4,29,60,50,20,5,70,174,180,100,30,6,169,490,609,
%T A367211 420,175,42,7,408,1352,1960,1624,840,280,56,8,985,3672,6084,5880,3654,
%U A367211 1512,420,72,9,2378,9850,18360,20280,14700,7308,2520,600,90,10
%N A367211 Triangular array read by rows: T(n, k) = binomial(n, k) * A000129(n - k) for 0 <= k < n.
%C A367211 T(n, k) are the coefficients of the polynomials p(1, x) = 1, p(2, x) = 2 + 2*x, p(n, x) = u*p(n-1, x) + v*p(n-2, x) for n >= 3, where u = p(2, x), v = 1 - 2*x - x^2.
%C A367211 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.
%H A367211 Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, <a href="http://math.colgate.edu/~integers/s14/s14.Abstract.html">Characterization of the strong divisibility property for generalized Fibonacci polynomials</a>, Integers, 18 (2018), Paper No. A14.
%F A367211 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 - 2*x - x^2.
%F A367211 p(n, x) = k*(b^n - c^n), where k = sqrt(1/8), b = x + 1 - sqrt(2), c = x + 1 + sqrt(2).
%F A367211 From _Werner Schulte_, Nov 24 2023 and Nov 25 2023: (Start)
%F A367211 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.
%F A367211 T(n, k) = T(n-1, k-1) * n / k for 0 < k < n and T(n, 0) = A000129(n) for n > 0.
%F A367211 T(n, k) = A000129(n-k) * binomial(n, k) for 0 <= k < n.
%F A367211 G.f.: t / (1 - (2+2*x) * t - (1-2*x-x^2) * t^2). (End)
%e A367211 First nine rows:
%e A367211 [n\k] 0 1 2 3 4 5 6 7 8
%e A367211 [1] 1;
%e A367211 [2] 2 2;
%e A367211 [3] 5 6 3;
%e A367211 [4] 12 20 12 4;
%e A367211 [5] 29 60 50 20 5;
%e A367211 [6] 70 174 180 100 30 6;
%e A367211 [7] 169 490 609 420 175 42 7;
%e A367211 [8] 408 1352 1960 1624 840 280 56 8;
%e A367211 [9] 985 3672 6084 5880 3654 1512 420 72 9;
%e A367211 .
%e A367211 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.
%p A367211 P := proc(n) option remember; ifelse(n <= 1, n, 2*P(n - 1) + P(n - 2)) end:
%p A367211 T := (n, k) -> P(n - k) * binomial(n, k):
%p A367211 for n from 1 to 9 do [n], seq(T(n, k), k = 0..n-1) od;
%p A367211 # (after _Werner Schulte_) _Peter Luschny_, Nov 24 2023
%t A367211 p[1, x_] := 1; p[2, x_] := 2 + 2 x; u[x_] := p[2, x]; v[x_] := 1 - 2 x - x^2;
%t A367211 p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
%t A367211 Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%t A367211 Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%t A367211 (* Or: *)
%t A367211 T[n_, k_] := Module[{P},
%t A367211 P[m_] := P[m] = If[m <= 1, m, 2*P[m - 1] + P[m - 2]];
%t A367211 P[n - k] * Binomial[n, k] ];
%t A367211 Table[T[n, k], {n, 1, 9}, {k, 0, n - 1}] (* _Peter Luschny_, Mar 07 2025 *)
%Y A367211 Cf. A000129 (column 1, Pell numbers), 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.
%K A367211 nonn,tabl
%O A367211 1,2
%A A367211 _Clark Kimberling_, Nov 13 2023
%E A367211 New name using a formula of _Werner Schulte_ by _Peter Luschny_, Mar 07 2025