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A232192 G.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^n - 1)^n. 2
1, 1, 1, 5, 44, 519, 7590, 132347, 2689046, 62644234, 1651650774, 48731341965, 1592908456996, 57173688136781, 2235773294509565, 94608603077007214, 4306708055122614542, 209823573154587335730, 10892496561736261641371, 600171728539156939466278 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..175

FORMULA

G.f. satisfies:

(1) A(x) = 1 + x*Sum_{n>=0} A(x)^(n^2) / (1 + A(x)^n)^(n+1). - Paul D. Hanna, Mar 31 2018

(2) A(x) = 1 + Series_Reversion(x/G(x))

(3) A(x) = 1 + x*G(A(x)-1)

where G(x) is the g.f. of A122400, the number of square (0,1)-matrices without zero rows and with exactly n entries equal to 1.

a(n) ~ c * d^n * n! / n^(3/2), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491726382774188556341627278..., r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0, and c = 0.12140554666... . - Vaclav Kotesovec, May 07 2014

EXAMPLE

G.f.: A(x) = 1 + x + x^2 + 5*x^3 + 44*x^4 + 519*x^5 + 7590*x^6 + 132347*x^7 + 2689046*x^8 + 62644234*x^9 + 1651650774*x^10 +...

where

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

Also,

A(x) = 1 + x/2  +  x*A(x)/(1 + A(x))^2  +  x*A(x)^4/(1 + A(x)^2)^3  +  x*A(x)^9/(1 + A(x)^3)^4  +  x*A(x)^16/(1 + A(x)^4)^5  +  x*A(x)^25/(1 + A(x)^5)^6  + ...

PROG

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*sum(m=0, n, (A^m-1+x*O(x^n))^m)); polcoeff(A, n)}

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

CROSSREFS

Cf. A122400, A192945, A192946, A192947, A192948.

Sequence in context: A054766 A252830 A301434 * A249791 A215648 A195242

Adjacent sequences:  A232189 A232190 A232191 * A232193 A232194 A232195

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 20 2013

STATUS

approved

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Last modified April 10 09:20 EDT 2021. Contains 342845 sequences. (Running on oeis4.)