A235321 E.g.f. satisfies: A'(x) = A(x)^4 * A(-x)^2 with A(0) = 1.

1, 1, 2, 10, 56, 496, 4592, 58240, 753536, 12428416, 206700032, 4194910720, 85291222016, 2054623707136, 49390297628672, 1376983620812800, 38210541166493696, 1209980100295622656, 38070418025153626112, 1349585751030491054080, 47479274652991633227776

Offset

0,3

Links

Table of n, a(n) for n=0..20.

Formula

E.g.f. A(x) satisfies: 3 - 15*A(x) + 20*A(x)^2 - (8 - 15*x)*A(x)^5 = 0.

E.g.f.: 1/(1 - Series_Reversion( Integral (1-x^2)^2 dx )).

E.g.f.: 1 + Series_Reversion( Integral (1+2*x)^2/(1+x)^6 dx ).

Recurrence: 4096*(100*n - 341)*a(n) = 30720*(25*n - 83)*a(n-1) + 640*(4500*n^3 - 35595*n^2 + 92875*n - 81419)*a(n-2) - 1800*(1500*n^3 - 13480*n^2 + 40050*n - 39221)*a(n-3) - 81*(5*n - 19)*(5*n - 18)*(5*n - 17)*(5*n - 16)*(100*n - 241)*a(n-4). - Vaclav Kotesovec, Jan 28 2014

a(n) ~ n! * (5/2)^(1/3) * (15/8)^n / (GAMMA(1/3) * n^(2/3)). - Vaclav Kotesovec, Jan 28 2014

Example

E.g.f.: A(x) = 1 + x + 2*x^2/2! + 10*x^3/3! + 56*x^4/4! + 496*x^5/5! +...
Related series.
A(x)^2 = 1 + 2*x + 6*x^2/2! + 32*x^3/3! + 216*x^4/4! + 1952*x^5/5! +...
A(x)^4 = 1 + 4*x + 20*x^2/2! + 136*x^3/3! + 1160*x^4/4! + 12064*x^5/5! +...
A(x)^5 = 1 + 5*x + 30*x^2/2! + 230*x^3/3! + 2160*x^4/4! + 24200*x^5/5! +...
where 3 - 15*A(x) + 20*A(x)^2 = (8 - 15*x)*A(x)^5.
Note that 1 - 1/A(x) is an odd function that begins:
1 - 1/A(x) = x + 4*x^3/3! + 136*x^5/5! + 12544*x^7/7! + 2240896*x^9/9! +...
where Series_Reversion(1 - 1/A(x)) = x - 2*x^3/3 + x^5/5.

Mathematica

CoefficientList[1/(1 - InverseSeries[Series[x - 2*x^3/3 + x^5/5, {x, 0, 20}], x]), x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 28 2014 *)

Prog

(PARI) /* By definition, A'(x) = A(x)^4 * A(-x)^2: */
{a(n)=local(A=1); for(i=0, n, A=1+intformal(A^4*subst(A, x, -x)^2 +x*O(x^n) )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* By equation: 3 - 15*A(x) + 20*A(x)^2 - (8 - 15*x)*A(x)^5 = 0 */
{a(n)=local(A=1); for(i=0, n, A=(3+20*A^2-(8-15*x)*A^5)/15 +x*O(x^n)); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* E.g.f. 1/(1 - Series_Reversion(Integral (1-x^2)^2 dx)): */
{a(n)=local(A=1); A=1/(1-serreverse(x-2*x^3/3+x^5/5 +x^2*O(x^n))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* E.g.f. 1 + Series_Reversion(Integral (1+2*x)^2/(1+x)^6 dx): */
{a(n)=local(A=1); A=1+serreverse( intformal((1+2*x)^2/(1+x +x^2*O(x^n))^6 )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A235322.

Sequence in context: A093303 A199163 A303830 * A364306 A371770 A369487

Adjacent sequences: A235318 A235319 A235320 * A235322 A235323 A235324

Keyword

nonn

Author

Paul D. Hanna, Jan 05 2014

Status

approved