A100097 -id:A100097 - cp's OEIS frontend

A100095 An inverse Chebyshev transform of the Fibonacci numbers.

0, 1, 1, 5, 7, 25, 41, 125, 225, 625, 1195, 3125, 6227, 15625, 32059, 78125, 163727, 390625, 831505, 1953125, 4206145, 9765625, 21215481, 48828125, 106782837, 244140625, 536618341, 1220703125, 2693492305, 6103515625, 13507578125

Offset: 0

Paul Barry, Nov 03 2004

Image of x/(1-x-x^2) under the transform g(x)->(1/sqrt(1-4*x^2))*g(x*c(x^2)), where c(x) is the g.f. of the Catalan numbers A000108. This is the inverse of the Chebyshev transform which takes A(x) to ((1-x^2)/(1+x^2))*A(x/(1+x^2)).
Hankel transform is (-1)^n*(2^n-0^n)/2. Hankel transform of a(n+1) is A141125. - Paul Barry, Jun 05 2008
Basically A000351 interleaved with A144635. - Peter Luschny, May 31 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A000351, A144635, A000045, A100096, A100097.

CoefficientList[Series[(x^2*Sqrt[1-4*x^2]+x*(1-4*x^2))/((1-4*x^2)*(1-5*x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)

a(n):=sum(4^(j)*binomial((n-1)/2,j),j,0,(n-1)/2); /* Vladimir Kruchinin, May 31 2014 */

G.f.: (x^2*sqrt(1-4*x^2)+x*(1-4*x^2))/((1-4*x^2)*(1-5*x^2)).
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*Fibonacci(n-2*k).
Conjecture: (-n+2)*a(n) +(-n+3)*a(n-1) +(9*n-22)*a(n-2) +(9*n-31)*a(n-3) +20*(-n+3)*a(n-4) +20*(-n+4)*a(n-5)=0. - R. J. Mathar, Nov 24 2012
Recurrence: (n-2)*a(n) = (9*n-22)*a(n-2) - 20*(n-3)*a(n-4). - Vaclav Kotesovec, Feb 12 2014
a(n) ~ 5^((n-1)/2). - Vaclav Kotesovec, Feb 12 2014
a(n) = sum(j=0..(n-1)/2, 4^(j)*binomial((n-1)/2,j)). - Vladimir Kruchinin, May 31 2014
a(2*n) = 5^n/sqrt(5) - 2^n * (2*n-1)!! * hypergeom([1, n+1/2], [n+1], 4/5)/(5*n!), a(2*n+1) = 5^n. - Vladimir Reshetnikov, Oct 13 2016

A100096 An inverse Chebyshev transform of the Jacobsthal numbers.

Original entry on oeis.org

0, 1, 1, 6, 9, 36, 66, 218, 449, 1332, 2946, 8196, 18954, 50688, 120576, 314586, 761889, 1957092, 4794426, 12194828, 30093854, 76067256, 188595276, 474810276, 1180734234, 2965094536, 7387570516, 18521858088, 46203981924, 115721310552

Offset: 0

Paul Barry, Nov 03 2004

Image of x/(1-x-2*x^2) under the transform g(x)->(1/sqrt(1-4*x^2))*g(x*c(x^2)), where c(x) is the g.f. of the Catalan numbers A000108. This is the inverse of the Chebyshev transform which takes A(x) to ((1-x^2)/(1+x^2))*A(x/(1+x^2)).

Hankel transform is (-1)^n*n. Hankel transform of a(n+1) is A141124. - Paul Barry, Jun 05 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A100095, A100097.

CoefficientList[Series[x*Sqrt[1-4*x^2]*(3*Sqrt[1-4*x^2]+2*x+1)/(2*(4*x^2-1)*(10*x^2+x-2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)

G.f.: x*sqrt(1-4*x^2)*(3*sqrt(1-4*x^2)+2*x+1)/(2*(4*x^2-1)*(10*x^2+x-2)).
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*A001045(n-2*k).
Conjecture: 2*(-n+1)*a(n) +(n+3)*a(n-1) +18*(n-2)*a(n-2) +4*(-n-2)*a(n-3) +40*(-n+3)*a(n-4)=0. - R. J. Mathar, Nov 24 2012
a(n) ~ (5/2)^n / 3. - Vaclav Kotesovec, Feb 12 2014

A194349 E.g.f.: -log( sqrt(1-x^2) - x ).

Original entry on oeis.org

1, 2, 5, 24, 129, 960, 7965, 80640, 903105, 11612160, 163451925, 2554675200, 43259364225, 797058662400, 15764670046125, 334764638208000, 7571150452490625, 182111963185152000, 4634731528895593125, 124564582818643968000

Offset: 1

Paul D. Hanna, Aug 21 2011

nonn

Compare e.g.f. to arccosh(x) = log(sqrt(x^2-1) + x).

			E.g.f.: A(x) = x + 2*x^2/2! + 5*x^3/3! + 24*x^4/4! + 129*x^5/5! + ...
where
exp(A(x)) = 1 + 2*(x/2) + 6*(x/2)^2 + 16*(x/2)^3 + 46*(x/2)^4 + 128*(x/2)^5 + ... + A098617(n)*(x/2)^n + ...

Cf. A098617, A100097.

With[{nn=30},Rest[CoefficientList[Series[-Log[Sqrt[1-x^2]-x],{x,0,nn}], x] Range[0,nn]!]] (* Harvey P. Dale, Dec 01 2011 *)

{a(n)=n!*polcoeff(-log(sqrt(1-x^2+x*O(x^n))-x),n)}

{A000129(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=if(n<1,0,sum(k=0,floor((n+1)/2),binomial(n+1, k)*A000129(n+1-2*k))*(n-1)!/2^n)}

a(2*n) = 2^n*(2*n-1)! for n>=1.
a(n) = A100097(n+1)*(n-1)!/2^n for n>=1.
a(n) = (n-1)!/2^n * Sum_{k=0..floor((n+1)/2)} C(n+1,k)*A000129(n+1-2*k) for n >= 1. [From a formula of Paul Barry in A100097]
E.g.f.: log( (sqrt(1-x^2) + x)/(1-2*x^2) ).

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A100095 An inverse Chebyshev transform of the Fibonacci numbers.

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A100096 An inverse Chebyshev transform of the Jacobsthal numbers.

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A194349 E.g.f.: -log( sqrt(1-x^2) - x ).

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