# Binomial transforms of Fib, etc.

creigh at o2online.de creigh at o2online.de
Thu Sep 23 03:31:56 CEST 2004

```Thanks for all your previous comments and help!

A030191(n)  + 2a(n) + A093129(n+2) = 4A093129(n+1)
(a(n)) = (1,0,-5,-25,-100,-375,-1375,-5000,-18125,-65625)

A030191, Name: Scaled Chebyschev U-polynomial evaluated at sqrt(5)/2
A030191(n-1) is 2nd binomial transform of Fib(n)
A093129, binomial transform of Fib(2n-1)

superseeker's T004, below). Please also note transformations T101 and
T018 [= A030191, from above ].

p.s. if there are any serious mistakes in here this will be the last posting I
shall ever send at three in the morning!

Sincerely,
Creighton

Report on [ 1,0,-5,-25,-100,-375,-1375,-5000,-18125,-65625]:

[snip]

TEST: APPLY VARIOUS TRANSFORMATIONS TO SEQUENCE AND LOOK IT
UP IN THE ENCYCLOPEDIA AGAIN

SUCCESS
(limited to 40 matches):

Transformation T004 gave a match with:
%I A093131
%S A093131 0,1,5,20,75,275,1000,3625,13125,47500,171875,621875,2250000,
[snip],
%N A093131 Binomial transform of Fib(2n).
%F A093131 G.f.: x/(1-5x+5x^2); a(n)=(((5+sqrt(5))/2)^n-((5-sqrt(5))/2)
^n)/sqrt(5); a(n)=A093130(n)/2^n.
%Y A093131 Cf. A000045.
[snip]
%A A093131 Paul Barry (pbarry(AT)wit.ie), Mar 23 2004

Transformation T101 gave a match with:
%I A098149
%S A098149 1,1,4,11,29,76,199,521,1364,3571,9349,24476,64079,167761,439204,
[snip]
%N A098149 a(n) = 2(a(n-2) - a(n-1)) + a(n-3), with a(0) = a(1) = -1 and
a(2) = 4.
%C A098149 Sequence relates bisections of Lucas and Fibonacci numbers.
%C A098149 2*a(n) + A098150(n) = 8*(-1)^(n+1)*A001519(n) - (-1)^(n+1)*A005248
(n+1).
[snip]
%A A098149 Creighton Dement (creigh(AT)o2online.de), Aug 29 2004
%E A098149 More terms from RGWv (rgwv(AT)rgwv.com), Sep 1 2004

Transformation T005 gave a match with:
%I A039717
%S A039717 1,4,15,55,200,725,2625,9500,34375,124375,450000,1628125,5890625,[snip]
%N A039717 Row sums of convolution triangle A030523.
[snip]
%A A039717 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)

Transformation T018 gave a match with:
%S A030191 1,5,20,75,275,1000,3625,13125,47500,171875,621875,2250000,8140625,
[snip]
%N A030191 Scaled Chebyshev U-polynomial evaluated at sqrt(5)/2.
[snip]
%F A030191 a(n)=(sqrt(5))^n*U(n,sqrt(5)/2), g.f.: 1/(5*(x^2-x+1/5)), a(2*k+1)
=5^(k+1)*F(2*k+2), F(n) = Fibonacci
(A000045), a(2*k)=5^k*L(2*k+1), L(n) = Lucas (A000032)
%F A030191 a(n-1)=sum(k=0,n,C(n,k)*F(2*k)) - Benoit Cloitre (abcloitre(AT)
%F A030191 a(n) = 5*a(n-1)-5*a(n-2). - Benoit Cloitre (abcloitre(AT)wanadoo.
fr), Oct 23 2003
%F A030191 a(n-1)=((5/2+sqrt(5)/2)^n-(5/2-sqrt(5)/2)^n)/sqrt(5) is the
2nd binomial transform of Fib(n), the first binomial
transform of Fib(2n), and its n-th term is the n-th term of the third binomial
transform of Fib(3n) divided by 2^n. - Paul
Barry (pbarry(AT)wit.ie), Mar 23 2004
[snip]
%A A030191 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)

%S A093131 0,1,5,20,75,275,1000,3625,13125,47500,171875,621875,2250000,8140625,
[snip]
%N A093131 Binomial transform of Fib(2n).
%F A093131 G.f.: x/(1-5x+5x^2); a(n)=(((5+sqrt(5))/2)^n-((5-sqrt(5))/2)
^n)/sqrt(5); a(n)=A093130(n)/2^n.
%Y A093131 Cf. A000045.
[snip]
%A A093131 Paul Barry (pbarry(AT)wit.ie), Mar 23 2004

Transformation T019 gave a match with:
%I A039717
%S A039717 1,4,15,55,200,725,2625,9500,34375,124375,450000,1628125,5890625, %N A039717
Row sums of convolution triangle A030523.
[snip]
%A A039717 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)

List of transformations used:
T004 sequence divided by the gcd of its elements, from the 2nd term
T005 sequence divided by the gcd of its elements, from the 3rd term
T018 sequence u[j+1]-u[j]
T019 sequence u[j+2]-2*u[j+1]+u[j]
T101 inverse binomial transform: b(n)=SUM (-1)^(n-k)*C(n,k)*a(k), k=0..
n

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