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A235369 E.g.f. satisfies: A(x) = 1/(1 - Integral A(x) + A(x)^2 dx). 1
1, 2, 14, 170, 2942, 66122, 1828094, 59999690, 2279713982, 98416537802, 4758057790334, 254660698294730, 14948094619714622, 954799973383805642, 65929683232042222334, 4893783473730361785290, 388585728666772137187262, 32866836871725315639639242, 2950043008427035823210644094 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Compare to: G(x) = 1/(1 - Integral G(x) dx) holds when G(x) is the e.g.f. of odd double factorials (A001147).

Compare to: G(x) = 1/(1 - Integral G(x)^2 dx) holds when G(x) is the e.g.f. of triple factorial numbers (A007559).

Compare to: G(x) = 1 + Integral G(x)+G(x)^2 dx holds when G(x) is the e.g.f. of A000629 (number of necklaces of partitions of n+1 labeled beads).

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

FORMULA

E.g.f. A(x) satisfies:

(1) A(x) = 1/(1 - 2*Series_Reversion( Integral (1-2*x)^2/(1-x) dx )).

(2) A(x) = 1/sqrt(1 - 4*Series_Reversion( Integral sqrt(1-4*x)*C(x) dx )), where C(x) = 1 + x*C(x)^2 = (1 - sqrt(1-4*x))/(2*x) is the Catalan function (A000108).

(3) A(x) = exp( Integral A(x)^2 + A(x)^3 dx ). - Paul D. Hanna, Aug 21 2016

(4) A(x) = 1 + Integral  A(x)^3 + A(x)^4 dx. - Paul D. Hanna, Aug 21 2016

a(n) ~ n! / (GAMMA(1/3) * 3^(1/3) * (log(2)-1/2)^(n+1/3) * n^(2/3)). - Vaclav Kotesovec, Jan 27 2014

EXAMPLE

E.g.f.: A(x) = 1 + 2*x + 14*x^2/2! + 170*x^3/3! + 2942*x^4/4! + 66122*x^5/5! +...

Related series.

A(x)^2 = 1 + 4*x + 36*x^2/2! + 508*x^3/3! + 9780*x^4/4! + 238684*x^5/5! +...

The series 1 - 1/A(x) = Integral A(x) + A(x)^2 dx and begins:

1 - 1/A(x) = 2*x + 6*x^2/2! + 50*x^3/3! + 678*x^4/4! + 12722*x^5/5! +...

where the series reversion of (1 - 1/A(x))/2 equals Integral (1-2*x)^2/(1-x) dx.

The series 1 - 1/A(x)^2 begins:

1 - 1/A(x)^2 = 4*x + 4*x^2/2! + 28*x^3/3! + 340*x^4/4! + 5884*x^5/5! +...

where the series reversion of (1 - 1/A(x)^2)/4 begins:

x - x^2/2 - 2*x^3/3 - 5*x^4/4 - 14*x^5/5 - 42*x^6/6 - 132*x^7/7 - 429*x^8/8 - 1430*x^9/9 +...+ -A000108(n)*x^(n+1)/(n+1) +...

Also, we have

A(x) = exp( Integral A(x)^2 + A(x)^3 dx ), and thus

A'(x) = A(x)^3 + A(x)^4, where

A(x)^3 = 1 + 6*x + 66*x^2/2! + 1062*x^3/3! + 22530*x^4/4! + 593766*x^5/5! + 18705666*x^6/6! + 685717542*x^7/7! + 28674765570*x^8/8! +...

A(x)^4 = 1 + 8*x + 104*x^2/2! + 1880*x^3/3! + 43592*x^4/4! + 1234328*x^5/5! + 41294024*x^6/6! + 1593996440*x^7/7! + 69741772232*x^8/8! +...

MATHEMATICA

CoefficientList[1/(1-2*InverseSeries[Series[-2*x^2-Log[-1+x]+I*Pi, {x, 0, 20}], x]), x]*Range[0, 20]! (* Vaclav Kotesovec, Jan 27 2014 *)

PROG

(PARI) {a(n) = my(A=1); for(i=1, n, A = 1/(1 - intformal(A + A^2+x*O(x^n)))); n!*polcoeff(A, n)}

for(n=0, 20, print1(a(n), ", "))

(PARI) {a(n) = my(A = 1/(1 - 2*serreverse( intformal((1-2*x)^2/(1-x +x*O(x^n)) )))); n!*polcoeff(A, n)}

for(n=0, 20, print1(a(n), ", "))

(PARI) {a(n) = my(A=1); for(i=1, n, A = exp( intformal(A^2+A^3 +x*O(x^n)) ) ); n!*polcoeff(A, n)}

for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Aug 21 2016

(PARI) {a(n) = my(A=1+x); for(i=1, n, A = 1 + intformal(A^3 + A^4 +x*O(x^n))); n!*polcoeff(A, n)}

for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Aug 21 2016

CROSSREFS

Cf. A228608.

Sequence in context: A046247 A141012 A351277 * A351274 A228476 A308449

Adjacent sequences:  A235366 A235367 A235368 * A235370 A235371 A235372

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 08 2014

STATUS

approved

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Last modified May 22 09:50 EDT 2022. Contains 353949 sequences. (Running on oeis4.)