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A337756
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G.f. A(x) satisfies: 1 = Sum_{n>=0} (n+1)*(n+2)/2! * 3^n * ((1+x)^n - A(x))^n.
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4
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1, 1, 6, 180, 7845, 434448, 28594494, 2157238350, 182404049175, 17026549342770, 1735705779016158, 191667825521201286, 22781050822806698709, 2899308092950790588988, 393385952195184523370994, 56691647586489579559334352, 8649001755741912766806347253, 1392791055204268736953260163092
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OFFSET
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0,3
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COMMENTS
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In general, 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 = 3 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} C(n+2,2) * 3^n * ( (1+x)^n - A(x) )^n.
(2) 1 = Sum_{n>=0} C(n+2,2) * 3^n * (1+x)^(n^2) / (1 + 3*(1+x)^n*A(x))^(n+3).
a(n) ~ c * (1 + 3*exp(1/r))^n * r^(2*n) * n! * n^(3/2), 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 = 0.00671991787239... - Vaclav Kotesovec, Oct 13 2020
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EXAMPLE
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G.f.: A(x) = 1 + x + 6*x^2 + 180*x^3 + 7845*x^4 + 434448*x^5 + 28594494*x^6 + 2157238350*x^7 + 182404049175*x^8 + ...
where
1 = 1 + 3*3*((1+x) - A(x)) + 6*3^2*((1+x)^2 - A(x))^2 + 10*3^3*((1+x)^3 - A(x))^3 + 15*3^4*((1+x)^4 - A(x))^4 + 21*3^5*((1+x)^5 - A(x))^5 + 28*3^6*((1+x)^6 - A(x))^6 + 38*3^7*((1+x)^7 - A(x))^7 + ... + C(n+2,2)*3^n*((1+x)^n - A(x))^n + ...
Also,
1 = 1/(1 + 3*A(x))^3 + 3*3*(1+x)/(1 + 3*(1+x)*A(x))^4 + 6*3^2*(1+x)^4/(1 + 3*(1+x)^2*A(x))^5 + 10*3^3*(1+x)^9/(1 + 3*(1+x)^3*A(x))^6 + 15*3^4*(1+x)^16/(1 + 3*(1+x)^4*A(x))^7 + 21*3^5*(1+x)^25/(1 + 3*(1+x)^5*A(x))^8 + 28*3^6*(1+x)^36/(1 + 3*(1+x)^6*A(x))^9 + ... + C(n+2,2)*3^n*(1+x)^(n^2)/(1 + 3*(1+x)^n*A(x))^(n+3) + ...
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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, (m+1)*(m+2)/2! * 3^m * ((1+x)^m - Ser(A))^m ) )[#A]/9 ); 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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