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A209704
Triangle of coefficients of polynomials v(n,x) jointly generated with A209703; see the Formula section.
3
1, 3, 1, 4, 3, 2, 5, 6, 8, 3, 6, 10, 18, 14, 5, 7, 15, 33, 38, 27, 8, 8, 21, 54, 81, 83, 49, 13, 9, 28, 82, 150, 197, 170, 89, 21, 10, 36, 118, 253, 401, 448, 342, 159, 34, 11, 45, 163, 399, 736, 999, 987, 671, 282, 55, 12, 55, 218, 598, 1253, 1988, 2387, 2106
OFFSET
1,2
COMMENTS
For n>1, row n starts with n+1, followed by the n-th
triangular number, and ends in F(n+1), where F=A000045
(Fibonacci numbers).
Column 3: A166830.
Row sums: A048654.
Alternating row sums: 1,2,3,4,5,6,7,8,9,...
For a discussion and guide to related arrays, see A208510.
FORMULA
u(n,x)=x*u(n-1,x)+x*v(n-1,x),
v(n,x)=(x+1)*u(n-1,x)+v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
EXAMPLE
First five rows:
1
3...1
4...3....2
5...6....8....3
6...10...18...14...5
First three polynomials v(n,x): 1, 3 + x , 4 + 3x + 2x^2.
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := (x + 1)*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[%] (* A209703 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209704 *)
CROSSREFS
Sequence in context: A104764 A152842 A307280 * A339106 A082909 A335906
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 12 2012
STATUS
approved