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 1-3 of 3 results.

A107086 G.f. A(x) satisfies: A(x)^4 = A(x^2)^2 + 4*x.

Original entry on oeis.org

1, 1, -1, 2, -5, 13, -35, 99, -289, 857, -2578, 7864, -24252, 75430, -236348, 745431, -2364399, 7536482, -24127482, 77544613, -250098478, 809169322, -2625483810, 8541037140, -27851360659, 91018956200, -298052119611, 977825373366, -3213513271929, 10577811289462, -34870732260397
Offset: 0

Views

Author

Paul D. Hanna, May 11 2005

Keywords

Comments

Self-convolution is A107087. Self-convolution 4th power is A107088.

Examples

			A(x)^4 = 1 + 4*x + 2*x^2 - x^4 + 2*x^6 - 5*x^8 + 12*x^10 - 30*x^12 +...
A(x^2)^2 = 1 + 2*x^2 - x^4 + 2*x^6 - 5*x^8 + 12*x^10 - 30*x^12 +...

Links

Crossrefs

Cf. A107087, A107088, A187814, A228711.

Programs

  • Mathematica
    nmin = 0; nmax = 30; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x]^4 - A[x^2]^2 - 4x + O[x]^(n+1), x][[2;;]] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 2, nmax}];
a /@ Range[nmin, nmax] /. sol (* Jean-François Alcover, Nov 07 2019 *)
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=(subst(A, x, x^2)^2+4*x+x*O(x^n))^(1/4)); polcoeff(A, n, x)}
for(n=0,40,print1(a(n),", "))

A223026 G.f. A(x) satisfies: A(x)^8 = A(x^2)^4 + 8*x.

Original entry on oeis.org

1, 1, -3, 14, -76, 441, -2678, 16813, -108093, 707451, -4696017, 31530792, -213715953, 1460072247, -10042361784, 69473047716, -483046768116, 3373552141194, -23653214175084, 166422650191122, -1174621198245837, 8314055808436788, -58998774106863513
Offset: 0

Views

Author

Paul D. Hanna, Mar 11 2013

Keywords

Comments

The limit a(n+1)/a(n) seems to be near -7.46...

Examples

			G.f.: A(x) = 1 + x - 3*x^2 + 14*x^3 - 76*x^4 + 441*x^5 - 2678*x^6 +-...
where
A(x)^8 = 1 + 8*x + 4*x^2 - 6*x^4 + 24*x^6 - 117*x^8 + 612*x^10 - 3426*x^12 +-...
A(x^2)^4 = 1 + 4*x^2 - 6*x^4 + 24*x^6 - 117*x^8 + 612*x^10 - 3426*x^12 +-...
A(x)^2 = 1 + 2*x - 5*x^2 + 22*x^3 - 115*x^4 + 646*x^5 - 3822*x^6 +-...

Links

Crossrefs

Cf. A107086, A107089, A228711, A228712.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, #binary(n), A=(subst(A, x, x^2)^4+8*x+x*O(x^n))^(1/8)); polcoeff(A, n, x)}
for(n=0, 20, print1(a(n), ", "))

Formula

Self-convolution yields A228711.

A274483 G.f. A(x) satisfies: A(x)^2 = A( x^2/(1 - 4*x - 2*x^2) ).

Original entry on oeis.org

1, 2, 8, 32, 138, 612, 2784, 12896, 60635, 288614, 1388104, 6735808, 32938438, 162156828, 803026176, 3997462368, 19991321445, 100387500906, 505950179016, 2558352514272, 12974595610122, 65975538192036, 336293496474144, 1717927441213152, 8793426613714734, 45092543870052092, 231621905868337424, 1191586088094887936, 6138909938284313524, 31668826322371245256, 163571372589617459584, 845826517521629901888, 4378463647900723645800
Offset: 1

Views

Author

Paul D. Hanna, Aug 03 2016

Keywords

Comments

Radius of convergence is r = (sqrt(33) - 5)/4 where A(r) = 1.
Compare the g.f. with the identities:
(1) F(x)^2 = F( x^2/(1 - 4*x + 6*x^2) ) when F(x) = x/(1-2*x).
(2) C(x)^2 = C( x^2/(1 - 4*x + 4*x^2) ) when C(x) = (1-2*x - sqrt(1-4*x))/(2*x) is a g.f. of the Catalan numbers (A000108).
More generally, if
F(x)^2 = F( x^2/(1 - 2*a*x + 2*(a^2 - b)*x^2) ),
then
F( x/(1 + a*x + b*x^2) )^2 = F( x^2/(1 + a^2*x^2 + b^2*x^4) ).

Examples

			G.f.: A(x) = x + 2*x^2 + 8*x^3 + 32*x^4 + 138*x^5 + 612*x^6 + 2784*x^7 + 12896*x^8 + 60635*x^9 + 288614*x^10 + 1388104*x^11 + 6735808*x^12 +...
such that A( x^2/(1 - 4*x - 2*x^2) ) = A(x)^2.
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 20*x^4 + 96*x^5 + 468*x^6 + 2288*x^7 + 11248*x^8 + 55552*x^9 + 275610*x^10 + 1373192*x^11 + 6869096*x^12 +...
The series reversion of g.f. A(x) begins
Series_Reversion(A(x)) = x - 2*x^2 + 8*x^4 - 10*x^5 - 24*x^6 + 64*x^7 + 64*x^8 - 327*x^9 - 172*x^10 + 1664*x^11 + 480*x^12 - 8858*x^13 - 1328*x^14 + 49344*x^15 + 3584*x^16 - 286432*x^17 - 9714*x^18 + 1723264*x^19 + 26800*x^20 - 10669788*x^21 - 73768*x^22 + 67557440*x^23 + 200448*x^24 +...
Now compare the expansion given by
x/Series_Reversion(A(x)) = 1 + 2*x + 4*x^2 - 6*x^4 + 24*x^6 - 117*x^8 + 612*x^10 - 3426*x^12 + 20184*x^14 - 122883*x^16 + 766464*x^18 - 4875378*x^20 + 31507728*x^22 - 206278686*x^24 + 1365201252*x^26 - 9118841784*x^28 + 61393574760*x^30 - 416193047280*x^32 + 2838492444204*x^34 +...
to the series G(x) such that G(x)^2 = G(x^2) + 8*x, which begins
G(x) = 1 + 4*x - 6*x^2 + 24*x^3 - 117*x^4 + 612*x^5 - 3426*x^6 + 20184*x^7 - 122883*x^8 + 766464*x^9 - 4875378*x^10 + 31507728*x^11 - 206278686*x^12 +...
and equals the square of the g.f. of A228711.

Crossrefs

Cf. A228711, A274484, A264224.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, #binary(n+1), A = sqrt( subst(A, x, x^2/(1-4*x-2*x^2 +x*O(x^n)) ) ) ); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) A( x/(1 + 2*x + 5*x^2) )^2 = A( x^2/(1 + 4*x^2 + 25*x^4) ).
(2) A(x) = -A( -x/(1 - 4*x) ).
(3) A( x/(1 + 2*x) ) = -A( -x/(1 - 2*x) ), an odd function.
(4) A( x/(1 + 2*x) )^2 = A( x^2/(1 - 6*x^2) ), an even function.
Given G(x) such that G(x)^2 = G(x^2) + 8*x, then g.f. A(x) satisfies:
(5) A(x) = x/(1-2*x) * G( A(x)^2 ),
(6) A(x) = Series_Reversion( x/(G(x)^2 - 6*x) ),
(7) G(x) = sqrt( x/Series_Reversion(A(x)) + 6*x ),
(8) G(x^2) = x/Series_Reversion(A(x)) - 2*x,
(9) A( x/(G(x)^2 - 6*x) ) = x,
(10) A( x/(G(x^2) + 2*x) ) = x,
(11) A(x)^2/(G(A(x)^4) + 2*A(x)^2) = x^2/(1 - 4*x - 2*x^2).
Sum_{k=0..n} binomial(n,k) * (-2)^(n-k) * a(k+1) = 0 for odd n.
Sum_{k=0..n} binomial(n,k) * (-4)^(n-k) * a(k+1) = (-1)^n * a(n+1) for n>=0.
Showing 1-3 of 3 results.

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