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A122070 Triangle T(n,k), 0<=k<=n, given by T(n,k)=Fibonacci(n+k+1)*binomial(n,k). 3
1, 1, 2, 2, 6, 5, 3, 15, 24, 13, 5, 32, 78, 84, 34, 8, 65, 210, 340, 275, 89, 13, 126, 510, 1100, 1335, 864, 233, 21, 238, 1155, 3115, 5040, 4893, 2639, 610, 34, 440, 2492, 8064, 16310, 21112, 17080, 7896, 1597, 55, 801, 5184, 19572, 47502, 78860, 82908, 57492, 23256, 4181 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Subtriangle of (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Mirror image of the triangle in A185384.

LINKS

Table of n, a(n) for n=0..54.

FORMULA

T(n,k) = A000045(n+k+1)*A007318(n,k) .

T(n,n) = Fibonacci(2*n+1) = A001519(n+1) .

Sum_{k, 0<=k<=n} T(n,k) = Fibonacci(3*n+1) = A033887(n) .

Sum_{k, 0<=k<=n}(-1)^k*T(n,k) =(-1)^n = A033999(n) .

Sum_{k, 0<=k<=[n/2]}T(n-k,k)=(Fibonacci(n+1))^2 = A007598(n+1).

Sum_{k, 0<=k<=n} T(n,k)*2^k = Fibonacci(4*n+1) = A033889(n).

Sum_{k, 0<=k<=n} T(n,k)^2 = A208588(n).

G.f.: (1-y*x)/(1-(1+3y)*x-(1+y-y^2)*x^2).

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

T(n,k) = A185384(n,n-k).

T(2n,n) = binomial(2n,n)*Fibonacci(3*n+1) = A208473(n).

EXAMPLE

Triangle begins:

1;

1, 2;

2, 6, 5;

3, 15, 24, 13;

5, 32, 78, 84, 34;

8, 65, 210, 340, 275, 89;

13, 126, 510, 1100, 1335, 864, 233;

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

1

0, 1

0, 1, 2

0, 2, 6, 5

0, 3, 15, 24, 13

0, 5, 32, 78, 84, 34

0, 8, 65, 210, 340, 275, 89

0, 13, 126, 510, 1100, 1335, 864, 233

CROSSREFS

Cf. A000045, A001519, A033887, A033889, A185384

Sequence in context: A019749 A209773 A209767 * A181661 A144160 A275142

Adjacent sequences:  A122067 A122068 A122069 * A122071 A122072 A122073

KEYWORD

nonn,tabl

AUTHOR

Philippe Deléham, Oct 15 2006, Mar 13 2012

EXTENSIONS

Corrected and extended by Philippe Deléham, Mar 13 2012

STATUS

approved

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Last modified April 25 12:19 EDT 2019. Contains 322456 sequences. (Running on oeis4.)