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A251699
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a(n) = (7*n+1) * (8*n+1)^(n-2) * 9^n.
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9
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1, 8, 1215, 400950, 207202941, 146509780644, 131737019154363, 143893722808861650, 185052548743241765625, 273919266332571877019712, 458736814135093804224189111, 857575304752878031562956215918, 1770298011965146072399475770453365, 3999656915702652258291935606835937500
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OFFSET
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0,2
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LINKS
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FORMULA
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Let G(x) = 1 + x*G(x)^9 be the g.f. of A062994, then the e.g.f. A(x) of this sequence satisfies:
(1) A(x) = exp( 9*x*A(x)^8 * G(x*A(x)^8)^8 ) / G(x*A(x)^8).
(2) A(x) = F(x*A(x)^8) where F(x) = exp(9*x*G(x)^8)/G(x) is the e.g.f. of A251669.
(3) A(x) = ( Series_Reversion( x*G(x)^8 / exp(72*x*G(x)^8) )/x )^(1/8).
E.g.f.: (-LambertW(-72*x)/(72*x))^(1/8) * (1 + LambertW(-72*x)/72). - Vaclav Kotesovec, Dec 07 2014
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EXAMPLE
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E.g.f.: A(x) = 1 + 8*x + 1215*x^2/2! + 400950*x^3/3! + 207202941*x^4/4! + 146509780644*x^5/5! +...
such that A(x) = exp( 9*x*A(x)^8 * G(x*A(x)^8)^8 ) / G(x*A(x)^8),
where G(x) = 1 + x*G(x)^9 is the g.f. A062994:
G(x) = 1 + x + 9*x^2 + 117*x^3 + 1785*x^4 + 29799*x^5 + 527085*x^6 +...
Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^8) where
F(x) = 1 + 8*x + 191*x^2/2! + 8310*x^3/3! + 537117*x^4/4! + 46444164*x^5/5! +...
F(x) = exp( 9*x*G(x)^8 ) / G(x) is the e.g.f. of A251669.
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MATHEMATICA
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Table[(7*n + 1)*(8*n + 1)^(n - 2)*9^n, {n, 0, 50}] (* G. C. Greubel, Nov 14 2017 *)
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PROG
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(PARI) {a(n) = (7*n+1) * (8*n+1)^(n-2) * 9^n}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=local(G=1, A=1); for(i=0, n, G = 1 + x*G^9 +x*O(x^n));
A = ( serreverse( x*G^8 / exp(72*x*G^8) )/x )^(1/8); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(Magma) [(7*n + 1)*(8*n + 1)^(n - 2)*9^n: n in [0..50]]; // G. C. Greubel, Nov 14 2017
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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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