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A209571 Triangle of coefficients of polynomials u(n,x) jointly generated with A209572; see the Formula section. 3
1, 1, 1, 1, 4, 1, 1, 4, 9, 1, 1, 4, 15, 16, 1, 1, 4, 15, 44, 25, 1, 1, 4, 15, 56, 105, 36, 1, 1, 4, 15, 56, 185, 216, 49, 1, 1, 4, 15, 56, 209, 524, 399, 64, 1, 1, 4, 15, 56, 209, 732, 1295, 680, 81, 1, 1, 4, 15, 56, 209, 780, 2303, 2864, 1089, 100, 1, 1, 4, 15, 56 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Penultimate number in row n is (n-1)^2, for n>1.

Combinatorial limit of row n satisfies linear recurrence

r(n)=4*r(n-1)-r(n-2) with r(1)=1 and r(2)=4. For a

discussion and guide to related arrays, see A208510.

LINKS

Table of n, a(n) for n=1..70.

FORMULA

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

v(n,x)=2x*u(n-1,x)+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

1...4....9....1

1...4....15...16...1

First three polynomials v(n,x): 1, 1 + x, 1 + 4x + x^2.

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 16;

u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];

v[n_, x_] := 2 x*u[n - 1, x] + 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[%]   (* A209571 *)

Table[Expand[v[n, x]], {n, 1, z}]

cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

TableForm[cv]

Flatten[%]   (* A209572 *)

CROSSREFS

Cf. A209572, A208510.

Sequence in context: A301626 A080061 A246595 * A269845 A124258 A001638

Adjacent sequences:  A209568 A209569 A209570 * A209572 A209573 A209574

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 11 2012

STATUS

approved

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Last modified October 18 05:31 EDT 2019. Contains 328146 sequences. (Running on oeis4.)