cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A193619 G.f. A(x) satisfies: A(x)^-2 + A(-x)^-2 = 2 and A(x)^2 - A(-x)^2 = -8*x.

Original entry on oeis.org

1, -2, 6, 12, -122, -316, 4124, 11608, -169018, -495724, 7676596, 23075112, -371737956, -1135805144, 18808209528, 58139400112, -982459035322, -3063548374604, 52579900855620, 165071778169864, -2868211199377740, -9053503669975944
Offset: 0

Views

Author

Paul D. Hanna, Aug 01 2011

Keywords

Examples

			G.f.: A(x) = 1 - 2*x + 6*x^2 + 12*x^3 - 122*x^4 - 316*x^5 + 4124*x^6 +...
where
A(x)^2 = 1 - 4*x + 16*x^2 - 256*x^4 + 8192*x^6 - 327680*x^8 +...
and
A(x)^-2 = 1 + 4*x - 64*x^3 + 2048*x^5 - 81920*x^7 + 3670016*x^9 +...

Links

Crossrefs

Cf. A193618.

Programs

  • PARI
    {a(n)=local(Ox=x*O(x^n),A=((sqrt(1+64*x^2+Ox)+1)*(sqrt(1+64*x^2+Ox)-8*x)/2)^(1/4));polcoeff(A,n)}
  • PARI
    N=40; x='x+O('x^N); Vec(sqrt((1-8*x+sqrt(1+64*x^2))/2)) \\ Seiichi Manyama, Aug 26 2020

Formula

G.f.: ( (sqrt(1+64*x^2) + 1)*(sqrt(1+64*x^2) - 8*x)/2 )^(1/4).
G.f. A(x) = 1/G(x) where G(x) is the g.f. of A193618.

A196864 G.f. A(x) satisfies: A(x)^3 + A(-x)^3 = 2 and A(x)^-3 - A(-x)^-3 = -18*x.

Original entry on oeis.org

1, 3, -9, -198, 1188, 30213, -220239, -5945238, 47541735, 1325876283, -11192990913, -318640183182, 2787445591416, 80483342059224, -722019579525288, -21063846331387728, 192542324985927324, 5661585516173303268, -52508399485861250604, -1553593208517770295816
Offset: 0

Views

Author

Paul D. Hanna, Oct 06 2011

Keywords

Examples

			 G.f.: A(x) = 1 + 3*x - 9*x^2 - 198*x^3 + 1188*x^4 + 30213*x^5 +...
where
A(x)^3 = 1 + 9*x - 729*x^3 + 118098*x^5 - 23914845*x^7 + 5423886846*x^9 +...
A(x)^-3 = 1 - 9*x + 81*x^2 - 6561*x^4 + 1062882*x^6 - 215233605*x^8 +...

Crossrefs

Cf. A196865, A193618, A193619, A196866, A196867, A196868, A196869.

Programs

  • PARI
    {a(n)=local(X=x+x*O(x^n));polcoeff((2*(sqrt(1+4*3^4*X^2) + 2*3^2*x)/(sqrt(1+4*3^4*X^2) + 1) )^(1/6),n)}

Formula

G.f.: ( 2*(sqrt(1+4*3^4*x^2) + 2*3^2*x)/(sqrt(1+4*3^4*x^2) + 1) )^(1/6).

A196865 G.f. A(x) satisfies: A(x)^-3 + A(-x)^-3 = 2 and A(x)^3 - A(-x)^3 = 18*x.

Original entry on oeis.org

1, 3, 18, -117, -1971, 16119, 343278, -3036528, -71818164, 661017348, 16593480504, -156436510221, -4080815440497, 39095628518637, 1047594828442626, -10152600834566916, -277489726161424569, 2712640349690579349, 75279129630178436622, -740885355955719640809
Offset: 0

Views

Author

Paul D. Hanna, Oct 06 2011

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 18*x^2 - 117*x^3 - 1971*x^4 + 16119*x^5 +...
where
A(x)^3 = 1 + 9*x + 81*x^2 - 6561*x^4 + 1062882*x^6 - 215233605*x^8 +...
A(x)^-3 = 1 - 9*x + 729*x^3 - 118098*x^5 + 23914845*x^7 - 5423886846*x^9 +...

Crossrefs

Cf. A196864, A193618, A193619, A196866, A196867, A196868, A196869.

Programs

  • PARI
    {a(n)=local(A=[1,3]);for(k=2,n,A=concat(A,0);if(k%2==1,A[#A]=-Vec(Ser(A)^3)[#A]/3,A[#A]=Vec(Ser(A)^-3)[#A]/3));A[n+1]}
  • PARI
    {a(n)=local(X=x+x*O(x^n));polcoeff(((sqrt(1+4*3^4*X^2) + 2*3^2*x)*(sqrt(1+4*3^4*X^2) + 1)/2 )^(1/6),n)}

Formula

G.f.: ( (sqrt(1+4*3^4*x^2) + 2*3^2*x)*(sqrt(1+4*3^4*x^2) + 1)/2 )^(1/6).

A196866 G.f. A(x) satisfies: A(x)^4 + A(-x)^4 = 2 and A(x)^-4 - A(-x)^-4 = -32*x.

Original entry on oeis.org

1, 4, -24, -800, 9824, 381824, -5715712, -236348416, 3885237760, 166141515776, -2884493168640, -125973507063808, 2266868356071424, 100441740460359680, -1853741093854511104, -83006642599731134464, 1561071322451916750848, 70464426394180291919872
Offset: 0

Views

Author

Paul D. Hanna, Oct 06 2011

Keywords

Examples

			 G.f.: A(x) = 1 + 4*x - 24*x^2 - 800*x^3 + 9824*x^4 + 381824*x^5 +...
where
A(x)^4 = 1 + 16*x - 4096*x^3 + 2097152*x^5 - 1342177280*x^7 +...
A(x)^-4 = 1 - 16*x + 256*x^2 - 65536*x^4 + 33554432*x^6 - 21474836480*x^8 +...

Crossrefs

Cf. A196867, A193618, A193619, A196864, A196865, A196868, A196869.

Programs

  • PARI
    {a(n)=local(X=x+x*O(x^n));polcoeff((2*(sqrt(1+4*4^4*X^2) + 2*4^2*x)/(sqrt(1+4*4^4*X^2) + 1) )^(1/8),n)}

Formula

G.f.: ( 2*(sqrt(1+4*4^4*x^2) + 2*4^2*x)/(sqrt(1+4*4^4*x^2) + 1) )^(1/8).

A196867 G.f. A(x) satisfies: A(x)^-4 + A(-x)^-4 = 2 and A(x)^4 - A(-x)^4 = 32*x.

Original entry on oeis.org

1, 4, 40, -544, -14240, 240512, 7905536, -144081920, -5248825856, 99459618816, 3842132979712, -74547398033408, -2991092285194240, 58965437254402048, 2429529032173420544, -48445417122664284160, -2035619492638819483648, 40941665274780773253120
Offset: 0

Views

Author

Paul D. Hanna, Oct 06 2011

Keywords

Examples

			 G.f.: A(x) = 1 + 4*x + 40*x^2 - 544*x^3 - 14240*x^4 + 240512*x^5 +...
where
A(x)^4 = 1 + 16*x + 256*x^2 - 65536*x^4 + 33554432*x^6 +...
A(x)^-4 = 1 - 16*x + 4096*x^3 - 2097152*x^5 + 1342177280*x^7 +...

Crossrefs

Cf. A196866, A193618, A193619, A196864, A196865, A196868, A196869.

Programs

  • PARI
    {a(n)=local(X=x+x*O(x^n));polcoeff(((sqrt(1+4*4^4*X^2) + 2*4^2*x)*(sqrt(1+4*4^4*X^2) + 1)/2 )^(1/8),n)}

Formula

G.f.: ( (sqrt(1+4*4^4*x^2) + 2*4^2*x)*(sqrt(1+4*4^4*x^2) + 1)/2 )^(1/8).

A196868 G.f. A(x) satisfies: A(x)^2 + A(-x)^2 = 2 and A(x)^3 - A(-x)^3 = 36*x.

Original entry on oeis.org

1, 6, -18, 144, -1026, 10368, -91044, 995328, -9630090, 109486080, -1120744188, 13042778112, -138540597588, 1637370298368, -17853248637000, 213325958873088, -2371846639850586, 28573129903177728, -322526246042905740, 3910007249908531200, -44670671340291807228
Offset: 0

Views

Author

Paul D. Hanna, Oct 06 2011

Keywords

Examples

			G.f.: A(x) = 1 + 6*x - 18*x^2 + 144*x^3 - 1026*x^4 + 10368*x^5 +...
where
A(x)^2 = 1 + 12*x + 72*x^3 + 3240*x^5 + 229392*x^7 + 20083464*x^9 +...
A(x)^3 = 1 + 18*x + 54*x^2 + 1134*x^4 + 63180*x^6 + 4903254*x^8 +...

Crossrefs

Cf. A196869, A193618, A193619, A196864, A196865, A196866, A196867.

Programs

  • PARI
    {a(n)=local(A=[1,6]);for(k=2,n,A=concat(A,0);if(k%2==0,A[#A]=-Vec(Ser(A)^2)[#A]/2,A[#A]=-Vec(Ser(A)^3)[#A]/3));A[n+1]}
  • PARI
    /* Using series reversion: */
{a(n) = my(A, B = 6*serreverse(x - 24*x^3 +x^2*O(x^n)) );
A = B + sqrt(1 - B^2); polcoeff(A,n)}
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Apr 04 2022

Formula

From Paul D. Hanna, Apr 04 2022: (Start)
a(2*n+1) = 6 * 24^n * binomial(3*n+1,n)/(3*n+1) for n >= 0.
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) (A(x)^2 + A(-x)^2)/2 = 1.
(2) (A(x)^3 - A(-x)^3)/2 = 18*x.
(3) (A(x) - A(-x))/2 = 6*Series_Reversion(x - 24*x^3).
(4) (A(x) + A(-x))/2 = sqrt( (1 + A(x)*A(-x))/2 ).
(5) A(x)*A(-x) = 1 - 72*Series_Reversion(x - 24*x^3)^2.
(6) A(x) = B(x) + sqrt(1 - B(x)^2), where B(x) = 6*Series_Reversion(x - 24*x^3).
(End)

A196869 G.f. A(x) satisfies: A(x)^3 + A(-x)^3 = 2 and A(x)^2 - A(-x)^2 = 24*x.

Original entry on oeis.org

1, 6, -36, 216, -2592, 23328, -311040, 3265920, -45349632, 517321728, -7336562688, 88159684608, -1266403590144, 15771513618432, -228509902503936, 2921050338066432, -42583086769766400, 555279063084564480, -8132204141176946688, 107718176292801085440
Offset: 0

Views

Author

Paul D. Hanna, Oct 06 2011

Keywords

Examples

			G.f.: A(x) = 1 + 6*x - 36*x^2 + 216*x^3 - 2592*x^4 + 23328*x^5 +...
where
A(x)^2 = 1 + 12*x - 36*x^2 - 1296*x^4 - 108864*x^6 - 12317184*x^8 +...
A(x)^3 = 1 + 18*x - 432*x^3 - 23328*x^5 - 2239488*x^7 - 272097792*x^9 +...

Crossrefs

Cf. A196868, A193618, A193619, A196864, A196865, A196866, A196867.

Programs

  • PARI
    {a(n)=local(A=[1,6]);for(k=2,n,A=concat(A,0);if(k%2==1,A[#A]=-Vec(Ser(A)^2)[#A]/2,A[#A]=-Vec(Ser(A)^3)[#A]/3));A[n+1]}

A246062 G.f.: sqrt( (1 + sqrt(1+8*x)) / (1 + sqrt(1-8*x)) ).

Original entry on oeis.org

1, 2, 2, 28, 54, 860, 2004, 33720, 86054, 1492908, 4019452, 71101832, 198310460, 3555617432, 10168382696, 184127171952, 536496907782, 9788598556876, 28937139277804, 531135371147368, 1588378827366868, 29295861148032584, 88439788292856856, 1637711104368641552
Offset: 0

Views

Author

Paul D. Hanna, Aug 24 2014

Keywords

Comments

Unsigned version of A193618.

Examples

			G.f.: A(x) = 1 + 2*x + 2*x^2 + 28*x^3 + 54*x^4 + 860*x^5 + 2004*x^6 +...

Links

Crossrefs

Cf. A193618, A000108.

Programs

  • Mathematica
    CoefficientList[Series[Sqrt[(1 + Sqrt[1 + 8*x])/(1 + Sqrt[1 - 8*x])], {x, 0, 20}], x] (* Vaclav Kotesovec, Aug 25 2014 *)
  • PARI
    {a(n)=polcoeff( sqrt((1+sqrt(1+8*x +O(x^(n+2))))/(1+sqrt(1-8*x +O(x^(n+2))))),n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f.: sqrt( C(2*x)/C(-2*x) ) where C(x) = 1 + x*C(x)^2 is the Catalan function (A000108).
G.f.: exp( Sum_{n>=1} binomial(4*n-2,2*n-1)/2 * (2*x)^(2*n-1)/(2*n-1) ).
G.f. satisfies: A(x)*A(-x) = 1.
a(n) ~ sqrt(sqrt(2)-1)*(1+sqrt(2)-(-1)^n) * 2^(3*n-2) / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 25 2014

A337397 Expansion of sqrt(2 / ( (1+64*x^2) * (1-8*x+sqrt(1+64*x^2)) )).

Original entry on oeis.org

1, 2, -34, -92, 1654, 4828, -88724, -268088, 4984486, 15361708, -287691196, -898052872, 16901635516, 53234639768, -1005474931816, -3187958034544, 60375963282182, 192405594166988, -3651655920615596, -11684176213422568, 222132094724096852, 713091439789994824, -13575872676384218776
Offset: 0

Views

Author

Seiichi Manyama, Aug 26 2020

Keywords

Links

Crossrefs

Column k=4 of A337464.
Cf. A002458, A193618, A337396.

Programs

  • Mathematica
    a[n_] := Sum[(-4)^(n - k) * Binomial[2*k, k] * Binomial[2*n + 1, 2*k], {k, 0, n}]; Array[a, 23, 0] (* Amiram Eldar, Aug 26 2020 *)
CoefficientList[Series[Sqrt[2/((1+64x^2)(1-8x+Sqrt[1+64x^2]))],{x,0,30}],x] (* Harvey P. Dale, Jul 24 2021 *)
  • PARI
    N=40; x='x+O('x^N); Vec(sqrt(2/((1+64*x^2)*(1-8*x+sqrt(1+64*x^2)))))
  • PARI
    {a(n) = sum(k=0, n, (-4)^(n-k)*binomial(2*k, k)*binomial(2*n+1, 2*k))}

Formula

a(n) = Sum_{k=0..n} (-4)^(n-k) * binomial(2*k,k) * binomial(2*n+1,2*k).
a(0) = 1, a(1) = 2 and n * (2*n+1) * (4*n-3) * a(n) = (4*n-1) * 2 * a(n-1) - 64 * (n-1) * (2*n-1) * (4*n+1) * a(n-2) for n > 1. - Seiichi Manyama, Aug 29 2020
Showing 1-9 of 9 results.

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