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A208335 Triangle of coefficients of polynomials v(n,x) jointly generated with A208834; see the Formula section. 4
1, 2, 1, 3, 3, 1, 4, 7, 5, 1, 5, 14, 15, 6, 1, 6, 25, 36, 23, 8, 1, 7, 41, 76, 69, 36, 9, 1, 8, 63, 147, 176, 123, 48, 11, 1, 9, 92, 266, 400, 355, 192, 66, 12, 1, 10, 129, 456, 834, 910, 635, 292, 82, 14, 1, 11, 175, 747, 1626, 2131, 1833, 1065, 410, 105, 15, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

row sums, u(n,1):  A000129

row sums, v(n,1):  A001333

alternating row sums, u(n,-1): 1,0,-1,-2,-3,-4,-5,-6,...

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

Subtriangle of the triangle T(n,k) given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 26 2012

LINKS

Table of n, a(n) for n=1..66.

FORMULA

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

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

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

From Philippe Deléham, Mar 26 2012: (Start)

As DELTA-triangle T(n,k) with 0 <= k <= n:

G.f.: (1-x+x^2-y^2*x^2)/(1-2*x+x^2-y*x^2-y^2*x^2).

T(n,k) = 2*T(n-1,k) - T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,1) = 1, T(2,0) = 2, T(1,1) = T(2,2) = 0. (End)

EXAMPLE

First five rows:

  1;

  2,  1;

  3,  3,  1;

  4,  7,  5,  1;

  5, 14, 15,  6,  1;

First five polynomials v(n,x):

  1

  2 +   x

  3 +  3x +   x^2

  4 +  7x +  5x^2 +  x^3

  5 + 14x + 15x^2 + 6x^3 + x^4

From Philippe Deléham, Mar 26 2012: (Start)

(1, 1, -1, 1, 0, 0, 0, ...) DELTA (0, 1, 0, -1, 0, 0, ...) begins:

  1;

  1,  0;

  2,  1,  0;

  3,  3,  1,  0;

  4,  7,  5,  1,  0;

  5, 14, 15,  6,  1,  0; (End)

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] + 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[%]  (* A208334 *)

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

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

TableForm[cv]

Flatten[%]  (* A208335  *)

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. A208334.

Sequence in context: A133804 A185943 A208337 * A208597 A179943 A089944

Adjacent sequences:  A208332 A208333 A208334 * A208336 A208337 A208338

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Feb 26 2012

STATUS

approved

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Last modified June 14 21:27 EDT 2021. Contains 345041 sequences. (Running on oeis4.)