OFFSET
1,2
COMMENTS
Column 1: 1,2,3,4,5,6,....... A000027
Column 2: 1,2,4,7,11,........ A000124
Column 3: 2,6,13,24,......... A003600
Last term in row n: 1,1,2,3,. A000045 (Fibonacci numbers)
Row sums: 1,3,7,17,41,...... A001333
Alternating row sums: 1,2,3,3,5,5,7,7,...; A109613
For a discussion and guide to related arrays, see A208510.
FORMULA
u(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x),
v(n,x)=x*u(n-1,x)+v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
EXAMPLE
First five rows:
1
2...1
3...2...2
4...4...6....3
5...7...13...11...5
First three polynomials v(n,x): 1, 2 + x , 3 + 2x + 2x^2.
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
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[%] (* A209755 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209756 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A133654 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A001333 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A033999 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A109613 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 14 2012
STATUS
approved