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A094441 Triangular array T(n,k)=F(n+1-k)C(n,k), 0<=k<=n. 10
1, 1, 1, 2, 2, 1, 3, 6, 3, 1, 5, 12, 12, 4, 1, 8, 25, 30, 20, 5, 1, 13, 48, 75, 60, 30, 6, 1, 21, 91, 168, 175, 105, 42, 7, 1, 34, 168, 364, 448, 350, 168, 56, 8, 1, 55, 306, 756, 1092, 1008, 630, 252, 72, 9, 1, 89, 550, 1530, 2520, 2730, 2016, 1050, 360, 90, 10, 1, 144 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Triangle of coefficients of polynomials u(n,x) jointly generated with A209415; see the Formula section.

Column 1:  Fibonacci numbers:  F(n)=A000045(n)

Column 2:  n*F(n)

Row sums:  odd-indexed Fibonacci numbers

Alternating row sums: signed Fibonacci numbers

Coefficient of x^n in u(n,x):  1

Coefficient of x^(n-1) in u(n,x):  n

Coefficient of x^(n-2) in u(n,x):  n(n+1)

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

Subtriangle of the triangle given by (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 27 2012

REFERENCES

E. KILIÇ, H, BELBACHIR, Generalized double binomial sums families by generating functions, http://ekilic.etu.edu.tr/list/DoubleSums2.pdf, 2014.

LINKS

Table of n, a(n) for n=1..67.

FORMULA

Sum_{k, 0<=k<=n} T(n,k)*x^k = A039834(n-1), A000045(n+1), A001519(n+1), A081567(n), A081568(n), A081569(n), A081570(n), A081571(n) for x = -1, 0, 1, 2, 3, 4, 5, 6 respectively. - Philippe Deléham, Dec 14 2009

From Clark Kimberling, Mar 09 2012: (Start)

A094441 shows the coefficient of the polynomials u(n,x) which are jointly generated with polynomials v(n,x) by these rules:

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

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

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

(End)

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-1) - T(n-2,k-2), T(1,0) = T(2,0) = T(2,1) = 1 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 27 2012

G.f. -(-1+x*y)/(1-2*x*y-x-x^2+x^2*y+x^2*y^2). - R. J. Mathar, Aug 11 2015

EXAMPLE

First five rows:

1

1...1

2...2....1

3...6....3....1

5...12...12...4...1

First three polynomials v(n,x): 1, 1 + x, 2 + 2x + x^2.

Contribution from Philippe Deléham, Mar 27 2012: (Start)

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

1

0, 1

0, 1, 1

0, 2, 2, 1

0, 3, 6, 3, 1

0, 5, 12, 12, 4, 1. (End)

MATHEMATICA

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

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

v[n_, x_] := 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[%]    (* A094441 *)

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

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

TableForm[cv]

Flatten[%]    (* A094442 *)

CROSSREFS

Cf. A094435, A094442, A000045.

Sequence in context: A134399 A094436 A286012 * A107230 A159830 A293472

Adjacent sequences:  A094438 A094439 A094440 * A094442 A094443 A094444

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, May 03 2004

STATUS

approved

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