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A337757
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G.f. A(x) satisfies: 1 = Sum_{n>=0} (n+1)*(n+2)*(n+3)/3! * 4^n * ((1+x)^n - A(x))^n.
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4
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1, 1, 10, 460, 30250, 2488776, 240707480, 26452491760, 3233941091480, 433611348176880, 63118887464611936, 9899442124162104960, 1662993951689377716800, 297806177944353392091200, 56626969607275080551099520, 11394470658417110387020266496, 2419172929237326590857901776560, 540511078482106447677809541679680
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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 = 4 with r = 4, 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+3,3) * 4^n * ( (1+x)^n - A(x) )^n.
(2) 1 = Sum_{n>=0} C(n+3,3) * 4^n * (1+x)^(n^2) / (1 + 4*(1+x)^n*A(x))^(n+4).
a(n) ~ c * (1 + 4*exp(1/r))^n * r^(2*n) * n! * n^(5/2), where r = 0.95894043087329419322124137165060249611787608513866855417024... is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/4 and c = 0.0012636042138... - Vaclav Kotesovec, Oct 13 2020
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EXAMPLE
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G.f.: A(x) = 1 + x + 10*x^2 + 460*x^3 + 30250*x^4 + 2488776*x^5 + 240707480*x^6 + 26452491760*x^7 + 3233941091480*x^8 + ...
where
1 = 1 + 4*4*((1+x) - A(x)) + 10*4^2*((1+x)^2 - A(x))^2 + 20*4^3*((1+x)^3 - A(x))^3 + 35*4^4*((1+x)^4 - A(x))^4 + 56*4^5*((1+x)^5 - A(x))^5 + 84*4^6*((1+x)^6 - A(x))^6 + 120*4^7*((1+x)^7 - A(x))^7 + ... + C(n+3,3)*4^n*((1+x)^n - A(x))^n + ...
Also,
1 = 1/(1 + 4*A(x))^4 + 4*4*(1+x)/(1 + 4*(1+x)*A(x))^5 + 10*4^2*(1+x)^4/(1 + (1+x)^2*A(x))^6 + 20*4^3*(1+x)^9/(1 + 4*(1+x)^3*A(x))^7 + 35*4^4*(1+x)^16/(1 + 4*(1+x)^4*A(x))^8 + 56*4^5*(1+x)^25/(1 + 4*(1+x)^5*A(x))^9 + 84*4^6*(1+x)^36/(1 + 4*(1+x)^6*A(x))^10 + ... + C(n+3,3)*4^n*(1+x)^(n^2)/(1 + 4*(1+x)^n*A(x))^(n+4) + ...
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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)*(m+3)/3! * 4^m * ((1+x)^m - Ser(A))^m ) )[#A]/16 ); 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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