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A209703 Triangle of coefficients of polynomials u(n,x) jointly generated with A209704; see the Formula section. 3
1, 0, 2, 0, 3, 3, 0, 4, 6, 5, 0, 5, 10, 14, 8, 0, 6, 15, 28, 28, 13, 0, 7, 21, 48, 66, 55, 21, 0, 8, 28, 75, 129, 149, 104, 34, 0, 9, 36, 110, 225, 326, 319, 193, 55, 0, 10, 45, 154, 363, 626, 774, 661, 352, 89, 0, 11, 55, 208, 553, 1099, 1625, 1761, 1332, 634 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
For n>1, row n begins with 0, has second term n, and ends with F(n+1), where F=A000045 (Fibonacci numbers); for n>2, column 2 consists of triangular numbers.
Row sums: A098790.
Alternating row sums: 1,-2,0,-3,-1,-4,-3,-5,-3,-6,,...
For a discussion and guide to related arrays, see A208510.
LINKS
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
0...2
0...3....3
0...4....6...5
0...5...10...14...8
First three polynomials v(n,x): 1, 2x, 3x + 3x^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: A265208 A265020 A325191 * A279779 A279587 A127952
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 12 2012
STATUS
approved

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Last modified July 13 14:24 EDT 2024. Contains 374284 sequences. (Running on oeis4.)