A208508 Triangle of coefficients of polynomials u(n,x) jointly generated with A208509; see the Formula section.

1, 1, 1, 1, 4, 1, 9, 1, 1, 16, 6, 1, 25, 20, 1, 1, 36, 50, 8, 1, 49, 105, 35, 1, 1, 64, 196, 112, 10, 1, 81, 336, 294, 54, 1, 1, 100, 540, 672, 210, 12, 1, 121, 825, 1386, 660, 77, 1, 1, 144, 1210, 2640, 1782, 352, 14, 1, 169, 1716, 4719, 4290, 1287, 104, 1, 1

Offset

1,5

Comments

col 1: A000012

col 2: A000290 (squares)

col 3: A002415

col 4: A040977

col 5: A054334

row sums, u(n,1): A083329

Links

Table of n, a(n) for n=1..64.

Formula

u(n,x)=u(n-1,x)+x*v(n-1,x),

v(n,x)=u(n-1,x)+v(n-1,x)+1,

where u(1,x)=1, v(1,x)=1.

Example

First five rows:
1
1...1
1...4
1...9....1
1...16...6
First five polynomials u(n,x):
1
1 + x
1 + 4x
1 + 9x + x^2
1 + 16x + 6x^2

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, 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[%]    (* A208508 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208509 *)

Crossrefs

Cf. A208509.

Sequence in context: A143469 A360610 A331147 * A123726 A323600 A336851

Adjacent sequences: A208505 A208506 A208507 * A208509 A208510 A208511

Keyword

nonn,tabf

Author

Clark Kimberling, Feb 27 2012

Status

approved