A128056 -id:A128056 - cp's OEIS frontend

A128053 a(n)=A128056(n)/A128055(n).

1, -3, -3, 7, 7, -9, -9, 47, 47, -123, -123, 161, 161, -843, -843, 2207, 2207, -2889, -2889, 15127, 15127, -39603, -39603, 51841, 51841, -271443, -271443, 710647, 710647, -930249, -930249, 4870847, 4870847, -12752043, -12752043

Offset: 0

Paul Barry, Feb 13 2007

Cf. A128052.

a(n)=(-1)^C(n+1,2)*(F(n-1)+F(n+1))(5/6-cos(pi*n/3)/3)(1+(-1)^n)/2+(F(n)+F(n+2))(5/6-cos(pi*(n+1)/3)/3)(1-(-1)^n)/2.

Empirical g.f.: -(x-1) * (x^10-2*x^9-5*x^8+2*x^7+9*x^6+9*x^4+2*x^3-5*x^2-2*x+1) / ((x^4+3*x^2+1)*(x^8-3*x^6+8*x^4-3*x^2+1)). - Colin Barker, Jun 27 2013

A128055 a(n) = 2^A128054(n).

Original entry on oeis.org

1, 1, 2, 4, 8, 32, 64, 64, 128, 256, 512, 2048, 4096, 4096, 8192, 16384, 32768, 131072, 262144, 262144, 524288, 1048576, 2097152, 8388608, 16777216, 16777216, 33554432, 67108864, 134217728, 536870912, 1073741824

Offset: 0

Paul Barry, Feb 13 2007

A factor in A128056.

The signed sequence 1,1,2,-4,-8,-32,-64,-64,-128,256,512... is the Hankel transform of the doubled sequence 1,1,2,2,6,6,... of central binomial coefficients (A128014). - Paul Barry, Sep 09 2008

Index entries for linear recurrences with constant coefficients, signature (2,-4,8,-16,32).

Cf. A128014, A128054, A128056.

LinearRecurrence[{2,-4,8,-16,32},{1,1,2,4,8},31] (* James C. McMahon, Jan 05 2025 *)

G.f.: (-1-16*x^4+4*x^3-4*x^2+x)/((2*x-1)*(4*x^2-2*x+1)*(4*x^2+2*x+1)). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009

Sum_{n>=0} 1/a(n) = 62/21. - Amiram Eldar, Mar 28 2023

A128057 Expansion of (1+x)/sqrt(1+4x^2).

Original entry on oeis.org

1, 1, -2, -2, 6, 6, -20, -20, 70, 70, -252, -252, 924, 924, -3432, -3432, 12870, 12870, -48620, -48620, 184756, 184756, -705432, -705432, 2704156, 2704156, -10400600, -10400600, 40116600, 40116600, -155117520, -155117520

Offset: 0

Paul Barry, Feb 13 2007

Hankel transform is A128056. Binomial transform is A128058. Unsigned version is A128014.

Cf. A000984.

CoefficientList[Series[(1+x)/Sqrt[1+4x^2],{x,0,40}],x] (* Harvey P. Dale, May 12 2015 *)

a(n)=(-1)^C(n,2)*(C(n,n/2)*(1+(-1)^n)/2+C(n-1,(n-1)/2)*(1-(-1)^n)/2);
Conjecture: n*(5*n-9)*a(n) +4*a(n-1) +4*(5*n-4)*(n-2)*a(n-2)=0. - R. J. Mathar, Dec 02 2014
D-finite with recurrence: n*a(n) +(n-2)*a(n-1) +4*(n-1)*a(n-2) +4*(n-3)*a(n-3)=0. - R. J. Mathar, Dec 02 2014

A128058 Expansion of 1/((1-x)sqrt(1-2x+5x^2)).

Original entry on oeis.org

1, 2, 1, -4, -9, 2, 43, 72, -53, -418, -549, 860, 4161, 4006, -11619, -41104, -24989, 145046, 399571, 89796, -1723259, -3787914, 829841, 19739016, 34642741, -26909998, -219300587, -300591148, 467328447, 2369124842

Offset: 0

Paul Barry, Feb 13 2007

Partial sums of A098331. Binomial transform of A128057. Hankel transform is A128056.

CoefficientList[Series[1/((1-x)Sqrt[1-2x+5x^2]),{x,0,50}],x] (* Harvey P. Dale, Jun 23 2018 *)

a(n)=sum{k=0..n, sum{j=0..k, C(k,j)*C(k-j,j)*(-1)^j}}

D-finite with recurrence: n*a(n) +(-3*n+1)*a(n-1) +(7*n-6)*a(n-2) +5*(-n+1)*a(n-3)=0. - R. J. Mathar, Jan 23 2020

A186335 A transform of the central binomial coefficients.

Original entry on oeis.org

1, 1, 4, 7, 21, 46, 127, 309, 832, 2131, 5709, 15010, 40281, 107423, 289314, 778087, 2103361, 5687938, 15427099, 41880357, 113912236, 310148223, 845598389, 2307657222, 6304306171, 17237338021, 47170965082, 129181447969, 354027263457, 970851960736, 2664008539017

Offset: 0

Paul Barry, Feb 18 2011

Hankel transform is (-1)^n*A128056(n).

A186335 := proc(n)
    add(add(binomial(k-j,n-k-j)*binomial(k,j)*A000984(n-k-j),j=0..n),k=0..n) ;
end proc: # R. J. Mathar, Feb 13 2015

Table[Sum[Sum[Binomial[k-j,n-k-j]*Binomial[k,j]*Binomial[2*(n-k-j),n-k-j], {j,0,n}], {k,0,n}], {n,0,40}] (* Vaclav Kotesovec, Oct 30 2017 *)

a(n)=sum{k=0..n, sum{j=0..n, binomial(k-j,n-k-j)*binomial(k,j)*A000984(n-k-j)}}.
Conjecture: n*a(n) +(-2*n+1)*a(n-1) +5*(-n+1)*a(n-2) +3*(2*n-3)*a(n-3) +5*(n-2)*a(n-4)=0. - R. J. Mathar, Feb 13 2015
a(n) ~ ((1+sqrt(21))/2)^(n + 3/2) / (2 * 21^(1/4) * sqrt(Pi*n)) . - Vaclav Kotesovec, Oct 30 2017

cp's OEIS Frontend

A128053 a(n)=A128056(n)/A128055(n).

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A128055 a(n) = 2^A128054(n).

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A128057 Expansion of (1+x)/sqrt(1+4x^2).

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A128058 Expansion of 1/((1-x)sqrt(1-2x+5x^2)).

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A186335 A transform of the central binomial coefficients.

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