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A245265
E.g.f. satisfies: A(x) = exp(x/(1-x*A(x)^4)).
5
1, 1, 3, 37, 649, 15461, 471571, 17456041, 760880625, 38178439849, 2167446089251, 137359883836781, 9612722107574521, 736277501363180557, 61265207586681046131, 5503291392884323494961, 530778414439201798454881, 54706967800114521799571921, 6000952913613549583603208515
OFFSET
0,3
COMMENTS
Generally, if e.g.f. satisfies: A(x) = exp(x/(1-x*A(x)^p)), p>=1, then
r = 4*LambertW(sqrt(p)/2)^2 / (p*(1+2*LambertW(sqrt(p)/2))),
A(r) = (sqrt(p)/(2*LambertW(sqrt(p)/2)))^(2/p),
a(n) ~ p^(n-1+1/p) * (1+2*LambertW(sqrt(p)/2))^(n+1/2) * n^(n-1) / (sqrt(1+LambertW(sqrt(p)/2)) * exp(n) * 2^(2*n+2/p) * LambertW(sqrt(p)/2)^(2*n+2/p-1/2)).
FORMULA
a(n) = Sum_{k=0..n} n! * (1 + 4*(n-k))^(k-1)/k! * C(n-1,n-k).
a(n) ~ n^(n-1) * (1+2*LambertW(1))^(n+1/2) / (exp(n) * (LambertW(1))^(2*n) * (4*sqrt(1+LambertW(1)))). - Vaclav Kotesovec, Jul 15 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 37*x^3/3! + 649*x^4/4! + 15461*x^5/5! + 471571*x^6/6! + ...
MATHEMATICA
Table[Sum[n! * (1 + 4*(n-k))^(k-1)/k! * Binomial[n-1, n-k], {k, 0, n}], {n, 0, 20}]
PROG
(PARI) for(n=0, 30, print1(sum(k=0, n, n!*(1 + 4*(n-k))^(k-1)/k!*binomial(n-1, n-k)), ", ")) \\ G. C. Greubel, Nov 17 2017
CROSSREFS
Cf. A161630 (p=1), A212722 (p=2), A212917 (p=3).
Cf. A030178.
Cf. A366234 (log).
Sequence in context: A362724 A333385 A331114 * A362672 A143639 A143412
KEYWORD
nonn,easy
AUTHOR
Vaclav Kotesovec, Jul 15 2014
STATUS
approved