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A232192
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G.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^n - 1)^n.
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2
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1, 1, 1, 5, 44, 519, 7590, 132347, 2689046, 62644234, 1651650774, 48731341965, 1592908456996, 57173688136781, 2235773294509565, 94608603077007214, 4306708055122614542, 209823573154587335730, 10892496561736261641371, 600171728539156939466278
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OFFSET
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0,4
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LINKS
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FORMULA
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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
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EXAMPLE
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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 + ...
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PROG
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(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), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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