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A251698
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a(n) = (6*n+1) * (7*n+1)^(n-2) * 8^n.
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9
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1, 7, 832, 214016, 86118400, 47393538048, 33160072265728, 28180480000000000, 28194546272924860416, 32466269569728810844160, 42295727044150128912891904, 61505801717703291002224115712, 98762474157744880353280000000000, 173565347832317233669371533581090816, 331360760866451564310212841997955235840
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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)^8 be the g.f. of A007556, then the e.g.f. A(x) of this sequence satisfies:
(1) A(x) = exp( 8*x*A(x)^7 * G(x*A(x)^7)^7 ) / G(x*A(x)^7).
(2) A(x) = F(x*A(x)^7) where F(x) = exp(8*x*G(x)^7)/G(x) is the e.g.f. of A251668.
(3) A(x) = ( Series_Reversion( x*G(x)^7 / exp(56*x*G(x)^7) )/x )^(1/7).
E.g.f.: (-LambertW(-56*x)/(56*x))^(1/7) * (1 + LambertW(-56*x)/56). - Vaclav Kotesovec, Dec 07 2014
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EXAMPLE
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E.g.f.: A(x) = 1 + 7*x + 832*x^2/2! + 214016*x^3/3! + 86118400*x^4/4! + 47393538048*x^5/5! +...
such that A(x) = exp( 8*x*A(x)^7 * G(x*A(x)^7)^7 ) / G(x*A(x)^7),
where G(x) = 1 + x*G(x)^8 is the g.f. A007556:
G(x) = 1 + x + 8*x^2 + 92*x^3 + 1240*x^4 + 18278*x^5 + 285384*x^6 +...
Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^7) where
F(x) = 1 + 7*x + 146*x^2/2! + 5570*x^3/3! + 316376*x^4/4! + 24070168*x^5/5! +...
F(x) = exp( 8*x*G(x)^7 ) / G(x) is the e.g.f. of A251668.
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MATHEMATICA
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Table[(6*n + 1)*(7*n + 1)^(n - 2)*8^n, {n, 0, 50}] (* G. C. Greubel, Nov 14 2017 *)
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PROG
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(PARI) {a(n) = (6*n+1) * (7*n+1)^(n-2) * 8^n}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=local(G=1, A=1); for(i=0, n, G = 1 + x*G^8 +x*O(x^n));
A = ( serreverse( x*G^7 / exp(56*x*G^7) )/x )^(1/7); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(Magma) [(6*n + 1)*(7*n + 1)^(n - 2)*8^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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