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A317339 G.f. A(x) satisfies: Sum_{n>=0} ( 1/A(x) - 1/(1+x)^n )^n  =  1. 6
1, 1, 1, 4, 26, 239, 2768, 38267, 611193, 11040954, 222241117, 4929304517, 119423079917, 3137864557135, 88884310756274, 2700439386780586, 87603920737623984, 3022626187893726774, 110534722263602544357, 4270777627515614565004, 173854104446646589718022, 7437462737558953036993295 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..200

FORMULA

G.f. A(x) satisfies:

(1) 1 = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^n )^n.

(2) A(x) = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(n+1) )^n.

(3) 1 = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(n+1) )^n / (1+x)^(n+1).

a(n) ~ n^n / (2^(log(2)/2 + 5/2) * sqrt(1-log(2)) * exp(n) * (log(2))^(2*n + 1)). - Vaclav Kotesovec, Aug 12 2018

EXAMPLE

G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 26*x^4 + 239*x^5 + 2768*x^6 + 38267*x^7 + 611193*x^8 + 11040954*x^9 + 222241117*x^10 + ...

such that

1 = 1  +  (1/A(x) - 1/(1+x))  +  (1/A(x) - 1/(1+x)^2)^2  +  (1/A(x) - 1/(1+x)^3)^3  +  (1/A(x) - 1/(1+x)^4)^4  +  (1/A(x) - 1/(1+x)^5)^5  +  (1/A(x) - 1/(1+x)^6)^6  +  (1/A(x) - 1/(1+x)^7)^7  +  (1/A(x) - 1/(1+x)^8)^8  + ...

Also,

A(x) = 1  +  (1/A(x) - 1/(1+x)^2)  +  (1/A(x) - 1/(1+x)^3)^2  +  (1/A(x) - 1/(1+x)^4)^3  +  (1/A(x) - 1/(1+x)^5)^4  +  (1/A(x) - 1/(1+x)^6)^5  +  (1/A(x) - 1/(1+x)^7)^6  +  (1/A(x) - 1/(1+x)^8)^7  +  (1/A(x) - 1/(1+x)^9)^8  + ...

PROG

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ( 1/Ser(A) - 1/(1+x)^(m+1) )^m ) )[#A]/2 ); A[n+1]}

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

CROSSREFS

Cf. A317801, A317802, A317803, A317349.

Sequence in context: A300698 A244524 A209923 * A304338 A115416 A302606

Adjacent sequences:  A317336 A317337 A317338 * A317340 A317341 A317342

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Aug 10 2018

STATUS

approved

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Last modified April 22 06:32 EDT 2019. Contains 322329 sequences. (Running on oeis4.)