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A326283
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G.f. A(x) satisfies: 1 = Sum_{n>=0} 3^n * ((1+x)^n - A(x))^n.
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4
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1, 1, 3, 60, 1839, 75006, 3756681, 221373108, 14946242445, 1135512643905, 95807726020377, 8887382719047009, 899076658872459384, 98527185014966618211, 11629941100277822632476, 1471364340776928876041874, 198658352816171170123764339, 28515062757712810453709971653, 4336489052405032386983807856510, 696570875522773511632334350536870
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OFFSET
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0,3
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COMMENTS
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More generally, the following sums are equal:
(1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,
(2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - r*p*q^n)^(n+k),
for any fixed integer k; here, k = 1 with r = 3, p = -A(x), q = (1+x).
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} 3^n * ((1+x)^n - A(x))^n.
(2) 1 = Sum_{n>=0} 3^n * (1+x)^(n^2) / (1 + 3*(1+x)^n*A(x))^(n+1).
a(n) ~ c * (1 + 3*exp(1/r))^n * r^(2*n) * n! / sqrt(n), where r = 0.947093169766093813913446822751643203941993193936... is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/3 and c = 0.04495001143628... - Vaclav Kotesovec, Oct 13 2020
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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2 + 60*x^3 + 1839*x^4 + 75006*x^5 + 3756681*x^6 + 221373108*x^7 + 14946242445*x^8 + 1135512643905*x^9 + 95807726020377*x^10 + ...
such that
1 = 1 + 3*((1+x) - A(x)) + 3^2*((1+x)^2 - A(x))^2 + 3^3*((1+x)^3 - A(x))^3 + 3^4*((1+x)^4 - A(x))^4 + 3^5*((1+x)^5 - A(x))^5 + 3^6*((1+x)^6 - A(x))^6 + 3^7*((1+x)^7 - A(x))^7 + ...
Also,
1 = 1/(1 + 3*A(x)) + 3*(1+x)/(1 + 3*(1+x)*A(x))^2 + 3^2*(1+x)^4/(1 + 3*(1+x)^2*A(x))^3 + 3^3*(1+x)^9/(1 + 3*(1+x)^3*A(x))^4 + 3^4*(1+x)^16/(1 + 3*(1+x)^4*A(x))^5 + 3^5*(1+x)^25/(1 + 3*(1+x)^5*A(x))^6 + 3^6*(1+x)^36/(1 + 3*(1+x)^6*A(x))^7 + ...
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, 3^m*((1+x)^m - Ser(A))^m ) )[#A]/3 ); A[n+1]}
for(n=0, 30, 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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