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A209757 Triangle of coefficients of polynomials v(n,x) jointly generated with A013609; see the Formula section. 2
1, 3, 2, 5, 8, 4, 7, 18, 20, 8, 9, 32, 56, 48, 16, 11, 50, 120, 160, 112, 32, 13, 72, 220, 400, 432, 256, 64, 15, 98, 364, 840, 1232, 1120, 576, 128, 17, 128, 560, 1568, 2912, 3584, 2816, 1280, 256, 19, 162, 816, 2688, 6048, 9408, 9984, 6912, 2816, 512 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

For a discussion and guide to related arrays, see A208510.

Subtriangle of the triangle given by (1, 2, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 24 2012

LINKS

Table of n, a(n) for n=1..55.

FORMULA

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

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

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

Contribution from Philippe Deléham, Mar 24 2012. (Start)

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

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

T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - 2*T(n-2,k-2), T(0,0) = T(1,0) = 1, T(1,1) = T(2,2) = 0, T(2,0) = 3, T(2,1) = 2 and T(n,k) = 0 if k<0 or if k>n.

T(n,k) = 2^k*binomial(n-1,k)*(2*n-k-1)/(k+1). (End)

From Peter Bala, Dec 21 2014: (Start)

Following remarks assume an offset of 0.

T(n,k) = 2^k * A110813(n,k).

Riordan array ( (1 + x)/(1 - x)^2 , 2*x/(1 - x) ).

exp(2*x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(2*x)*(7 + 18*x + 20*x^2/2! + 8*x^3/3!) = 7 + 32*x + 120*x^2/2! + 400*x^3/3! + 1232*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), 2*x/(1 - x) ). (End)

EXAMPLE

First five rows:

1

3...2

5...8....4

7...18...20...8

9...32...56...48...16

First three polynomials v(n,x): 1, 3 + 2x , 5 + 8x + 4x^2.

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

1

1, 0

3, 2, 0

5, 8, 4, 0

7, 18, 20, 8, 0

9, 32, 56, 48, 16, 0. - Philippe Deléham, Mar 24 2012

MATHEMATICA

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

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

v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*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[%]    (* A013609 *)

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

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

TableForm[cv]

Flatten[%]    (* A209757 *)

CROSSREFS

Cf. A013609, A208510, A110813.

Sequence in context: A132776 A249741 A246275 * A208932 A189951 A209776

Adjacent sequences:  A209754 A209755 A209756 * A209758 A209759 A209760

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 23 2012

STATUS

approved

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Last modified October 17 19:24 EDT 2019. Contains 328127 sequences. (Running on oeis4.)