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A208340 Triangle of coefficients of polynomials v(n,x) jointly generated with A202390; see the Formula section. 4
1, 2, 2, 3, 6, 3, 4, 13, 14, 5, 5, 24, 41, 30, 8, 6, 40, 96, 109, 60, 13, 7, 62, 196, 308, 262, 116, 21, 8, 91, 364, 743, 868, 590, 218, 34, 9, 128, 630, 1604, 2413, 2240, 1267, 402, 55, 10, 174, 1032, 3186, 5926, 7046, 5424, 2627, 730, 89, 11, 230, 1617 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

v(n,n) = F(n+1), where F=A000045, the Fibonacci numbers

alternating row sums of v:  (1,0,0,0,0,0,0,0,...)

As triangle T(n,k) with 0<=k<=n, it is (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938 .

LINKS

Table of n, a(n) for n=1..58.

FORMULA

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

Contribution from Philippe Deléham, Feb 28 2012. (Start)

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

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

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

Sum_{k, 0<=k<=n} T(n,k)*x^k = A000007(n), A000027(n+1), A003946(n), A109115(n), A180031(n) for x = -1, 0, 1, 2, 3 respectively. (End)

EXAMPLE

First five rows:

1

2...2

3...6....3

4...13...14...5

5...24...41...30...8

The first five polynomials v(n,x):

1

2 + 2x

3 + 6x + 3x^2

4 + 13x + 14x^2 + 5x^3

5 + 24x + 41x^2 + 30x^3 + 8x^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 + 1)*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[%]    (* A202390 *)

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

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

TableForm[cv]

Flatten[%]    (* A208340 *)

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. A202390.

Sequence in context: A075196 A196912 A197079 * A308503 A196967 A210859

Adjacent sequences:  A208337 A208338 A208339 * A208341 A208342 A208343

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Feb 27 2012

STATUS

approved

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Last modified October 14 07:31 EDT 2019. Contains 327995 sequences. (Running on oeis4.)