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A207636 Triangle of coefficients of polynomials v(n,x) jointly generated with A207635; see Formula section. 3
1, 3, 2, 6, 7, 2, 12, 20, 11, 2, 24, 52, 42, 15, 2, 48, 128, 136, 72, 19, 2, 96, 304, 400, 280, 110, 23, 2, 192, 704, 1104, 960, 500, 156, 27, 2, 384, 1600, 2912, 3024, 1960, 812, 210, 31, 2, 768, 3584, 7424, 8960, 6944, 3584, 1232, 272, 35, 2, 1536, 7936 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

As triangle T(n,k) with 0<=k<=n, it is (3, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 26 2012

LINKS

Table of n, a(n) for n=1..57.

FORMULA

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

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.

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

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

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

Sum_{k, 0<=k<=n} T(n,k)*x^k = A003945(n), |A084244(n)|, A189274(n) for x = 0, 1, 3 respectively.

Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A040000(n), |A084244(n)|, A128625(n) for x = 0, 1, 2 respectively. (END) - Philippe Deléham, Feb 26 2012

EXAMPLE

First five rows:

1

3....2

6....7....2

12...20...11...2

24...52...42...15...2

MATHEMATICA

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

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

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

Table[Factor[u[n, x]], {n, 1, z}]

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

cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

TableForm[cu]

Flatten[%]  (* A207635 *)

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

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

TableForm[cv]

Flatten[%]  (* A207636 *)

CROSSREFS

Cf. A207635.

Cf. A084938, A003945, A040000

Sequence in context: A102004 A233208 A196518 * A125764 A023897 A267100

Adjacent sequences:  A207633 A207634 A207635 * A207637 A207638 A207639

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Feb 24 2012

STATUS

approved

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Last modified October 17 16:51 EDT 2019. Contains 328120 sequences. (Running on oeis4.)