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A208338 Triangle of coefficients of polynomials u(n,x) jointly generated with A208339; see the Formula section. 3
1, 1, 1, 1, 2, 3, 1, 3, 7, 7, 1, 4, 12, 20, 17, 1, 5, 18, 40, 57, 41, 1, 6, 25, 68, 129, 158, 99, 1, 7, 33, 105, 243, 399, 431, 239, 1, 8, 42, 152, 410, 824, 1200, 1160, 577, 1, 9, 52, 210, 642, 1506, 2692, 3528, 3089, 1393, 1, 10, 63, 280, 952, 2532, 5290 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Subtriangle of the triangle given by (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 09 2012

LINKS

Table of n, a(n) for n=1..62.

FORMULA

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

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

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

From Philippe Deléham, Apr 09 2012: (Start)

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

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

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)

EXAMPLE

First five rows:

  1;

  1,  1;

  1,  2,  3;

  1,  3,  7,  7;

  1,  4, 12, 20, 17;

First five polynomials u(n,x):

  1

  1 +  x

  1 + 2x +  3x^2

  1 + 3x +  7x^2 +  7x^3

  1 + 4x + 12x^2 + 20x^3 + 17x^4

From Philippe Deléham, Apr 09 2012: (Start)

(1, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, ...) begins:

  1;

  1,  0;

  1,  1,  0;

  1,  2,  3,  0;

  1,  3,  7,  7,  0;

  1,  4, 12, 20, 17,  0;

  1,  5, 18, 40, 57, 41,  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] + 2 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[%]   (* A208338 *)

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

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

TableForm[cv]

Flatten[%]   (* A208339 *)

CROSSREFS

Cf. A208339.

Sequence in context: A063967 A059397 A209567 * A236918 A152821 A071943

Adjacent sequences:  A208335 A208336 A208337 * A208339 A208340 A208341

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Feb 27 2012

STATUS

approved

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Last modified April 16 18:53 EDT 2021. Contains 343050 sequences. (Running on oeis4.)