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A193923 Triangular array:  the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=(x+1)^n and q(n,x)=sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers). 2
1, 1, 1, 1, 2, 3, 1, 3, 5, 8, 1, 4, 8, 13, 21, 1, 5, 12, 21, 34, 55, 1, 6, 17, 33, 55, 89, 144, 1, 7, 23, 50, 88, 144, 233, 377, 1, 8, 30, 73, 138, 232, 377, 610, 987, 1, 9, 38, 103, 211, 370, 609, 987, 1597, 2584, 1, 10, 47, 141, 314, 581, 979, 1596, 2584, 4181 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

The row sums equal A079289(2*n). - Johannes W. Meijer, Aug 12 2013

LINKS

Table of n, a(n) for n=0..64.

FORMULA

T(n,k) = sum(binomial(n+k-p-1, p), p=0..k). - Johannes W. Meijer, Aug 12 2013

EXAMPLE

First six rows:

1

1...1

1...2...3

1...3...5....8

1...4...8....13...21

1...5...12...21...34...55

MAPLE

T := proc(n, k) option remember: if k = 0 then return(1) fi: if k = n then return(combinat[fibonacci](2*n)) fi: T(n, k) := T(n-1, k-1) + T(n-1, k) end: seq(seq(T(n, k), k=0..n), n=0..9); # Johannes W. Meijer, Aug 12 2013

MATHEMATICA

p[n_, x_] := (x + 1)^n;

q[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];

t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;

w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1

g[n_] := CoefficientList[w[n, x], {x}]

TableForm[Table[Reverse[g[n]], {n, -1, z}]]

Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193923 *)

TableForm[Table[g[n], {n, -1, z}]]

Flatten[Table[g[n], {n, -1, z}]]  (* A193924 *)

CROSSREFS

Cf. A193722, A193924.

Sequence in context: A194740 A194762 A054250 * A198811 A067337 A180091

Adjacent sequences:  A193920 A193921 A193922 * A193924 A193925 A193926

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Aug 09 2011

STATUS

approved

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Last modified May 29 16:43 EDT 2020. Contains 334704 sequences. (Running on oeis4.)