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 1-4 of 4 results.

A258901 E.g.f. satisfies: A(x) = Integral 1 + A(x)^4 dx.

Original entry on oeis.org

1, 24, 32256, 285272064, 8967114326016, 735868743566229504, 130778914961055994085376, 44390350317502907443360825344, 26290393222157669992962395876622336, 25377887922329300948014930852183837507584, 37855568618678541873143615775486954119570128896
Offset: 0

Views

Author

Paul D. Hanna, Jun 14 2015

Keywords

Examples

			E.g.f.: A(x) = x + 24*x^5/5! + 32256*x^9/9! + 285272064*x^13/13! + 8967114326016*x^17/17! + 735868743566229504*x^21/21! +...
where Series_Reversion(A(x)) = x - x^5/5 + x^9/9 - x^13/13 + x^17/17 +...
Also, A(x) = S(x)/C(x) where
S(x) = x - 6*x^5/5! - 1764*x^9/9! - 7700616*x^13/13! - 147910405104*x^17/17! - 8310698364852576*x^21/21! +...+ A258900(n)*x^(4*n+1)/(4*n+1)! +...
C(x) = 1 - 6*x^4/4! - 1764*x^8/8! - 7700616*x^12/12! - 147910405104*x^16/16! - 8310698364852576*x^20/20! +...+ A258900(n)*x^(4*n)/(4*n)! +...
such that C(x)^4 + S(x)^4 = 1.

Links

Crossrefs

Cf. A000182, A000831, A258900, A258880, A258925, A258927, A259112, A259113, A258970.

Programs

  • Mathematica
    nmax=20; Table[CoefficientList[InverseSeries[Series[Integrate[1/(1+x^4),x],{x,0,4*nmax+1}],x],x][[4*n-2]] * (4*n-3)!, {n,1,nmax+1}] (* Vaclav Kotesovec, Jun 18 2015 *)
  • PARI
    /* E.g.f. Series_Reversion( Integral 1/(1+x^4) dx ): */
{a(n) = local(A=x); A = serreverse( intformal( 1/(1 + x^4 + O(x^(4*n+2))) ) ); (4*n+1)!*polcoeff(A,4*n+1)}
for(n=0,20,print1(a(n),", "))
  • PARI
    /* E.g.f. A(x) = Integral 1 + A(x)^4 dx.: */
{a(n) = local(A=x); for(i=1,n+1, A = intformal( 1 + A^4 + O(x^(4*n+2)) )); (4*n+1)!*polcoeff(A,4*n+1)}
for(n=0,20,print1(a(n),", "))

Formula

E.g.f. A(x) satisfies:
(1) A(x) = Series_Reversion( Integral 1/(1+x^4) dx ).
(2) A(x) = sqrt( tan( 2 * Integral A(x) dx ) ).
Let C(x) = S'(x) such that S(x) = Series_Reversion( Integral 1/(1-x^4)^(1/4) dx ) is the e.g.f. of A258900, then e.g.f. A(x) of this sequence satisfies:
(3) A(x) = S(x)/C(x),
(4) A(x) = Integral 1/C(x)^4 dx,
(5) A(x)^2 = S(x)^2/C(x)^2 = tan( 2 * Integral S(x)/C(x) dx ).
a(n) ~ 2^(6*n + 14/3) * (4*n)! * n^(1/3) / (3^(1/3) * Gamma(1/3) * Pi^(4*n + 4/3)). - Vaclav Kotesovec, Jun 18 2015

A258969 E.g.f.: A'(x) = 1 + A(x)^3, with A(0)=1.

Original entry on oeis.org

1, 2, 6, 42, 390, 4698, 69174, 1203498, 24163110, 549811962, 13982486166, 393026414922, 12099527531910, 404881353252378, 14632253175107574, 567974815524008298, 23567351945550373350, 1040985881615266375482, 48767788927611416600406, 2415210691383917131432842
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 15 2015

Keywords

Comments

Conjecture: A227250(n+1) = a(n).

Examples

			A(x) = 1 + 2*x + 6*x^2/2! + 42*x^3/3! + 390*x^4/4! + 4698*x^5/5! + ...
A'(x) = 2 + 6*x + 21*x^2 + 65*x^3 + 783*x^4/4 + 11529*x^5/20 + ...
1 + A(x)^3 = 2 + 6*x + 21*x^2 + 65*x^3 + 783*x^4/4 + 11529*x^5/20 + ...

Links

Crossrefs

Cf. A000831, A024396, A193534, A227250, A258970, A258971, A258880, A258994.

Programs

  • Mathematica
    nmax=20; Subscript[a,0]=1; egf=Sum[Subscript[a,k]*x^k, {k,0,nmax+1}]; Table[Subscript[a,k]*k!, {k,0,nmax}] /.Solve[Take[CoefficientList[Expand[1+egf^3-D[egf,x]],x],nmax]==ConstantArray[0,nmax]][[1]]
  • PARI
    {a(n) = local(A=1); A = 1 + serreverse( intformal( 1/((2+x)*(1+x+x^2) +x*O(x^n)) )); n!*polcoeff(A,n)}
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Jun 16 2015

Formula

a(n) ~ (3/(Pi/sqrt(3)-log(2)))^(n+1/2) * n^n / exp(n).
E.g.f.: 1 + Series_Reversion( Integral 1/((2+x)*(1+x+x^2)) dx ). - Paul D. Hanna, Jun 16 2015

A258994 E.g.f.: A'(x) = 1 + A(x)^6, with A(0)=1.

Original entry on oeis.org

1, 2, 12, 192, 4272, 124992, 4531392, 195869952, 9832326912, 562125837312, 36056880110592, 2564230500421632, 200237330428342272, 17032391106795159552, 1567547894591436275712, 155196096043697480466432, 16447362605632117421309952, 1857733260790463501532659712
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 16 2015

Keywords

Comments

In general, for k>1, if e.g.f. satisfies A'(x) = 1 + A(x)^k, with A(0)=1, then a(n) ~ n! * d^(n + 1/(k-1)) / ((k-1)^(1/(k-1)) * Gamma(1/(k-1)) * n^(1-1/(k-1))), where d = 1 / Sum_{j>=1} (-1)^(j+1)/(k*j-1).

Examples

			A(x) = 1 + 2*x + 12*x^2/2! + 192*x^3/3! + 4272*x^4/4! + 124992*x^5/5! + ...
A'(x) = 2 + 12*x + 96*x^2 + 712*x^3 + 5208*x^4 + 188808*x^5/5 + ...
1 + A(x)^6 = 2 + 12*x + 96*x^2 + 712*x^3 + 5208*x^4 + 188808*x^5/5 + ...

Links

Crossrefs

Cf. A000831 (k=2), A258969 (k=3), A258970 (k=4), A258971 (k=5), A258927.

Programs

  • Mathematica
    nmax=20; Subscript[a,0]=1; egf=Sum[Subscript[a,k]*x^k,{k,0,nmax+1}]; Table[Subscript[a,k]*k!,{k,0,nmax}] /.Solve[Take[CoefficientList[Expand[1+egf^6-D[egf,x]],x],nmax]==ConstantArray[0,nmax]][[1]]
  • PARI
    {a(n) = local(A=1); A = 1 + serreverse( intformal( 1/(1 + (1+x)^6 +x*O(x^n)) )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Jun 16 2015

Formula

a(n) ~ n! * d^(n+1/5) / (5^(1/5) * Gamma(1/5) * n^(4/5)), where d = 1 / Sum_{j>=1} (-1)^(j+1)/(6*j-1) = 6/(Pi - sqrt(3)*log(2+sqrt(3))) = 6.97224737278326506475991855023425659249063565...
E.g.f.: 1 + Series_Reversion( Integral 1/(1 + (1+x)^6) dx ). - Paul D. Hanna, Jun 16 2015

A258971 E.g.f.: A'(x) = 1 + A(x)^5, with A(0)=1.

Original entry on oeis.org

1, 2, 10, 130, 2330, 54770, 1591690, 55065250, 2209888250, 100922263250, 5167670934250, 293215490277250, 18260340583516250, 1238269550334211250, 90824251513716786250, 7164531681653318001250, 604824006980892825496250, 54406894886223009690031250
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 15 2015

Keywords

Comments

In general, for k>1, if e.g.f. satisfies A'(x) = 1 + A(x)^k, with A(0)=1, then a(n) ~ n! * d^(n + 1/(k-1)) / ((k-1)^(1/(k-1)) * Gamma(1/(k-1)) * n^(1-1/(k-1))), where d = 1 / Sum_{j>=1} (-1)^(j+1)/(k*j-1).

Examples

			A(x) = 1 + 2*x + 10*x^2/2! + 130*x^3/3! + 2330*x^4/4! + 54770*x^5/5! + ...
A'(x) = 2 + 10*x + 65*x^2 + 1165*x^3/3 + 27385*x^4/12 + 159169*x^5/12 + ...
1 + A(x)^5 = 2 + 10*x + 65*x^2 + 1165*x^3/3 + 27385*x^4/12 + 159169*x^5/12 + ...

Links

Crossrefs

Cf. A000831 (k=2), A258969 (k=3), A258970 (k=4), A258994 (k=6), A258925.

Programs

  • Mathematica
    nmax=20; Subscript[a,0]=1; egf=Sum[Subscript[a,k]*x^k, {k,0,nmax+1}]; Table[Subscript[a,k]*k!, {k,0,nmax}] /.Solve[Take[CoefficientList[Expand[1+egf^5-D[egf,x]],x],nmax]==ConstantArray[0,nmax]][[1]]
  • PARI
    {a(n) = local(A=1); A = 1 + serreverse( intformal( 1/(1 + (1+x)^5 +x*O(x^n)) )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Jun 16 2015

Formula

a(n) ~ n! * d^(n+1/4) / (4^(1/4) * Gamma(1/4) * n^(3/4)), where d = 1 / Sum_{j>=1} (-1)^(j+1)/(5*j-1) = 40*sqrt(5-sqrt(5)) / (8*sqrt(2)*Pi + sqrt(5+sqrt(5)) * ((9-5*sqrt(5))*log(2) + (sqrt(5)-5)*log(7+3*sqrt(5)))) = 5.53569595526739362969262739469167643400611216649309306882558956...
E.g.f.: 1 + Series_Reversion( Integral 1/(1 + (1+x)^5) dx ). - Paul D. Hanna, Jun 16 2015
Showing 1-4 of 4 results.

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