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A322738 E.g.f. A(x) = (1 + Integral A(x) dx) * (1 + Integral A(x)^2 dx). 2
1, 2, 8, 50, 422, 4480, 57300, 857364, 14690244, 283594200, 6090223440, 144002872968, 3717346949880, 104024775376416, 3136618299654000, 101380000924630416, 3496607473494821136, 128180947344040558752, 4976894571781037789184, 204030008190766804890912, 8806691099474138713650528 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Compare: G(x) = (1 + Integral G(x) dx)^2 holds when G(x) = 1/(1 - x)^2.

Compare: G(x) = (1 + Integral G(x)^2 dx)^2 holds when G(x) = 1/(1 - 3*x)^(2/3), the e.g.f. of the triple factorials product_{k=0..n-1} (3*k+2).

Compare: G(x) = (1 + Integral G(x)^m dx)^2 holds when G(x) = 1/(1 - (2*m-1)*x)^(2/(2*m-1)) = Sum_{n>=0} x^n/n! * product_{k=0..n-1} ((2*m-1)*k + 2).

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..397

FORMULA

E.g.f. A(x) satisfies the following relations.

(1) A(x) = (1 + Integral A(x) dx) * (1 + Integral A(x)^2 dx).

(2) A'(x) = A(x) * (1 + Integral A(x)^2 dx) + A(x)^2 * (1 + Integral A(x) dx).

(3) log(A(x)) = Integral [ A(x)/(1 + Integral A(x) dx) + A(x)^2/(1 + Integral A(x)^2 dx) ] dx.

(4a) log(1 + Integral A(x) dx) = Integral (1 + Integral A(x)^2 dx) dx.

(4b) log(1 + Integral A(x)^2 dx) = Integral A(x)*(1 + Integral A(x) dx) dx.

EXAMPLE

E.g.f.: A(x) = 1 + 2*x + 8*x^2/2! + 50*x^3/3! + 422*x^4/4! + 4480*x^5/5! + 57300*x^6/6! + 857364*x^7/7! + 14690244*x^8/8! + 283594200*x^9/9! + 6090223440*x^10/10! + ...

RELATED SERIES.

A(x)^2 = 1 + 4*x + 24*x^2/2! + 196*x^3/3! + 2028*x^4/4! + 25400*x^5/5! + 373400*x^6/6! + 6301408*x^7/7! + 120040416*x^8/8! + 2547619968*x^9/9! + ...

log(A(x)) = 2*x + 4*x^2/2! + 18*x^3/3! + 118*x^4/4! + 1028*x^5/5! + 11180*x^6/6! + 145784*x^7/7! + 2216600*x^8/8! + 38502688*x^9/9! + 752186400*x^10/10! + ...

such that

log(A(x)) = Integral [ (1 + Integral A(x)^2 dx) + A(x)*(1 + Integral A(x) dx) ] dx.

PROG

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

for(n=0, 25, print1(a(n), ", "))

CROSSREFS

Sequence in context: A000557 A193352 A002801 * A233436 A225052 A295759

Adjacent sequences:  A322735 A322736 A322737 * A322739 A322740 A322741

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 14 2019

STATUS

approved

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Last modified August 13 22:57 EDT 2020. Contains 336473 sequences. (Running on oeis4.)