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A355153
G.f. A(x) satisfies: 0 = Sum_{n=-oo..+oo} x^(n*(n+1)/2) * (x^n - (-1)^n*2*A(x))^(n+1).
2
1, 2, 10, 49, 272, 1617, 10082, 65101, 431635, 2921557, 20104828, 140240820, 989366180, 7046832503, 50604822586, 365995915453, 2663552184585, 19490777248544, 143322830835474, 1058514890796268, 7848460747315854, 58400364116985559, 435963301942052908
OFFSET
0,2
LINKS
FORMULA
G.f. A(x) satisfies:
(1) 0 = Sum_{n=-oo..+oo} x^(n*(n+1)/2) * (x^n - (-1)^n*2*A(x))^(n+1).
(2) 0 = Sum_{n=-oo..+oo} x^(3*n*(n-1)/2) / (1 - (-x)^n*2*A(x))^(n-1).
a(n) ~ c * d^n / n^(3/2), where d = 7.9902080716257105993518109688307894114113... and c = 0.651281200395360554284404869149552977212. - Vaclav Kotesovec, Jun 22 2022
EXAMPLE
G.f.: A(x) = 1 + 2*x + 10*x^2 + 49*x^3 + 272*x^4 + 1617*x^5 + 10082*x^6 + 65101*x^7 + 431635*x^8 + 2921557*x^9 + 20104828*x^10 + ...
where
0 = ... + x^6/(x^(-4) - 2*A(x))^3 + x^3/(x^(-3) + 2*A(x))^2 + x/(x^(-2) - 2*A(x)) + 1 + (1 - 2*A(x)) + x*(x + 2*A(x))^2 + x^3*(x^2 - 2*A(x))^3 + x^6*(x^3 + 2*A(x))^4 +--+ ...
PROG
(PARI) {a(n) = my(A=[1], t); for(i=1, n, A=concat(A, 0); t = ceil(sqrt(2*n+1));
A[#A] = polcoeff( sum(n=-t, t, x^(n*(n+1)/2) * (x^n - 2*(-1)^n*Ser(A))^(n+1) ), #A-1)/2); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Cf. A355152.
Sequence in context: A052576 A279852 A219662 * A268108 A143147 A317111
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 21 2022
STATUS
approved