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%I #15 Sep 08 2022 08:46:10
%S 1,7,832,214016,86118400,47393538048,33160072265728,28180480000000000,
%T 28194546272924860416,32466269569728810844160,
%U 42295727044150128912891904,61505801717703291002224115712,98762474157744880353280000000000,173565347832317233669371533581090816,331360760866451564310212841997955235840
%N a(n) = (6*n+1) * (7*n+1)^(n-2) * 8^n.
%H G. C. Greubel, <a href="/A251698/b251698.txt">Table of n, a(n) for n = 0..240</a>
%F 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:
%F (1) A(x) = exp( 8*x*A(x)^7 * G(x*A(x)^7)^7 ) / G(x*A(x)^7).
%F (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.
%F (3) A(x) = ( Series_Reversion( x*G(x)^7 / exp(56*x*G(x)^7) )/x )^(1/7).
%F E.g.f.: (-LambertW(-56*x)/(56*x))^(1/7) * (1 + LambertW(-56*x)/56). - _Vaclav Kotesovec_, Dec 07 2014
%e E.g.f.: A(x) = 1 + 7*x + 832*x^2/2! + 214016*x^3/3! + 86118400*x^4/4! + 47393538048*x^5/5! +...
%e such that A(x) = exp( 8*x*A(x)^7 * G(x*A(x)^7)^7 ) / G(x*A(x)^7),
%e where G(x) = 1 + x*G(x)^8 is the g.f. A007556:
%e G(x) = 1 + x + 8*x^2 + 92*x^3 + 1240*x^4 + 18278*x^5 + 285384*x^6 +...
%e Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^7) where
%e F(x) = 1 + 7*x + 146*x^2/2! + 5570*x^3/3! + 316376*x^4/4! + 24070168*x^5/5! +...
%e F(x) = exp( 8*x*G(x)^7 ) / G(x) is the e.g.f. of A251668.
%t Table[(6*n + 1)*(7*n + 1)^(n - 2)*8^n, {n, 0, 50}] (* _G. C. Greubel_, Nov 14 2017 *)
%o (PARI) {a(n) = (6*n+1) * (7*n+1)^(n-2) * 8^n}
%o for(n=0,20,print1(a(n),", "))
%o (PARI) {a(n)=local(G=1,A=1); for(i=0, n, G = 1 + x*G^8 +x*O(x^n));
%o A = ( serreverse( x*G^7 / exp(56*x*G^7) )/x )^(1/7); n!*polcoeff(A, n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (Magma) [(6*n + 1)*(7*n + 1)^(n - 2)*8^n: n in [0..50]]; // _G. C. Greubel_, Nov 14 2017
%Y Cf. A251668, A007556.
%Y Cf. Variants: A127670, A251693, A251694, A251695, A251696, A251697, A251699, A251700.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Dec 07 2014
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