[go: up one dir, main page]

login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

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
OFFSET
0,4
LINKS
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 20 2013
STATUS
approved