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A208336 Triangle of coefficients of polynomials u(n,x) jointly generated with A208337; see the Formula section. 5
1, 1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 4, 9, 10, 5, 1, 5, 14, 22, 20, 8, 1, 6, 20, 40, 51, 38, 13, 1, 7, 27, 65, 105, 111, 71, 21, 1, 8, 35, 98, 190, 256, 233, 130, 34, 1, 9, 44, 140, 315, 511, 594, 474, 235, 55, 1, 10, 54, 192, 490, 924, 1295, 1324, 942, 420, 89, 1, 11 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

coef. of x^(n-1) in u(n,x): A000045(n), Fibonacci numbers

coef. of x^(n-1) in v(n,x): A000045(n+1)

row sums, u(n,1):  A000129

row sums, v(n,1):  A001333

alternating row sums, u(n,-1): 1,0,1,0,1,0,1,0,1,0,...

alternating row sums, v(n,-1): 1,-1,1,-1,1,-1,1,-1,...

LINKS

Table of n, a(n) for n=1..68.

FORMULA

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

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

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

T(n,k) = A038137(n-1,k). - Philippe Deléham, Apr 05 2012

EXAMPLE

First five rows:

1

1...1

1...2...2

1...3...5...3

1...4...9...10...5

First five polynomials u(n,x):

1

1 + x

1 + 2x + 2x^2

1 + 3x + 5x^2 + 3x^3

1 + 4x + 9x^2 + 10x^3 + 5x^4

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 13;

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

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

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[%]    (* A208336 *)

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

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

TableForm[cv]

Flatten[%]    (* A208337 *)

Table[u[n, x] /. x -> 1, {n, 1, z}] (* u row sums *)

Table[v[n, x] /. x -> 1, {n, 1, z}] (* v row sums *)

Table[u[n, x] /. x -> -1, {n, 1, z}](* u alt. row sums *)

Table[v[n, x] /. x -> -1, {n, 1, z}](* v alt. row sums *)

CROSSREFS

Cf. A038137, A208337.

Sequence in context: A239830 A140767 A060850 * A038137 A073133 A106179

Adjacent sequences:  A208333 A208334 A208335 * A208337 A208338 A208339

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Feb 26 2012

STATUS

approved

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Last modified October 17 06:08 EDT 2019. Contains 328106 sequences. (Running on oeis4.)