A185081 -id:A185081 - cp's OEIS frontend

A209138 Triangle of coefficients of polynomials v(n,x) jointly generated with A209137; see the Formula section.

1, 1, 2, 2, 4, 3, 3, 9, 10, 5, 5, 18, 28, 22, 8, 8, 35, 68, 74, 45, 13, 13, 66, 154, 210, 177, 88, 21, 21, 122, 331, 541, 574, 397, 167, 34, 34, 222, 686, 1302, 1656, 1446, 850, 310, 55, 55, 399, 1382, 2982, 4404, 4614, 3434, 1758, 566, 89, 89, 710, 2723

Offset: 1

Clark Kimberling, Mar 05 2012

Every row begins and ends with a Fibonacci number (A000045).
u(n,1) = n-th row sum = 3^(n-1).
Alternating row sums: 1,-1,1,-1,1,-1,1,-1,1,-1,...
For a discussion and guide to related arrays, see A208510.

			First five rows:
  1;
  1,  2;
  2,  4,  3;
  3,  9, 10,  5;
  5, 18, 28, 22,  8;
First three polynomials v(n,x): 1, 1 + 2x, 2 + 4x + 3x^2.
From _Philippe Deléham_, Apr 11 2012: (Start)
Triangle in A185081 begins:
  1;
  0,  1;
  0,  1,  2;
  0,  2,  4,  3;
  0,  3,  9, 10,  5;
  0,  5, 18, 28, 22,  8;
  ... (End)

Cf. A209137, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := (x + 1)*u[n - 1, x] + x*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A209137 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209138 *)

u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
v(n,x) = (x+1)*u(n-1,x) + x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Apr 11 2012: (Start)
T(n,k) = A185081(n,k+1).
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), T(1,0) = T(2,0) = 1, T(2,1) = 2 and T(n,k) = 0 if k < 0 or if k >= n. (End)

A228815 Symmetric triangle, read by rows, related to Fibonacci numbers.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 5, 5, 2, 3, 10, 14, 10, 3, 5, 20, 36, 36, 20, 5, 8, 38, 83, 106, 83, 38, 8, 13, 71, 182, 281, 281, 182, 71, 13, 21, 130, 382, 690, 834, 690, 382, 130, 21, 34, 235, 778, 1606, 2268, 2268, 1606, 778, 235, 34, 55, 420, 1546, 3586, 5780, 6750

Offset: 0

Philippe Deléham, Oct 30 2013

Triangles satisfying the same recurrence: A091533, A091562, A185081, A205575, A209137, A209138.

			Triangle begins :
0
1, 1
1, 2, 1
2, 5, 5, 2
3, 10, 14, 10, 3
5, 20, 36, 36, 20, 5
8, 38, 83, 106, 83, 38, 8
13, 71, 182, 281, 281, 182, 71, 13
21, 130, 382, 690, 834, 690, 382, 130, 21
34, 235, 778, 1606, 2268, 2268, 1606, 778, 235, 34
55, 420, 1546, 3586, 5780, 6750, 5780, 3586, 1546, 420, 55

Cf. A000045 (1st column), A001629 (2nd column), A008998, A152011, A261055 (3rd column).

G.f.: x*(1+y)/(1-x-x*y-x^2-x^2*y-x^2*y^2).
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = 0, T(1,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n.
Sum_{k = 0..n} T(n,k)*x^k = A000045(n), 2*A015518(n), 3*A015524(n), 4*A200069(n) for x = 0, 1, 2, 3 respectively.
Sum_{k = 0..floor(n/2)} T(n-k,k) = A008998(n+1).

cp's OEIS Frontend

A209138 Triangle of coefficients of polynomials v(n,x) jointly generated with A209137; see the Formula section.

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