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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

Table of n, a(n) for n=1..65.

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

Cf. A209704, A208510.

Sequence in context: A265208 A265020 A325191 * A279779 A279587 A127952

Adjacent sequences:  A209700 A209701 A209702 * A209704 A209705 A209706

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 12 2012

STATUS

approved

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Last modified October 22 18:31 EDT 2019. Contains 328319 sequences. (Running on oeis4.)