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%I #10 Sep 08 2022 08:46:10
%S 1,6,539,104272,31513125,13018130762,6835288192159,4358439870247764,
%T 3271482918202092041,2826044644022395468750,2761781119675422226696419,
%U 3012587650584028093856586776,3628565076873134344787430377389,4783177086109789054912470697687698,6849486554475843842876951982177734375
%N a(n) = (5*n+1) * (6*n+1)^(n-2) * 7^n.
%H G. C. Greubel, <a href="/A251697/b251697.txt">Table of n, a(n) for n = 0..248</a>
%F Let G(x) = 1 + x*G(x)^7 be the g.f. of A002296, then the e.g.f. A(x) of this sequence satisfies:
%F (1) A(x) = exp( 7*x*A(x)^6 * G(x*A(x)^6)^6 ) / G(x*A(x)^6).
%F (2) A(x) = F(x*A(x)^6) where F(x) = exp(7*x*G(x)^6)/G(x) is the e.g.f. of A251667.
%F (3) A(x) = ( Series_Reversion( x*G(x)^6 / exp(42*x*G(x)^6) )/x )^(1/6).
%F E.g.f.: (-LambertW(-42*x)/(42*x))^(1/6) * (1 + LambertW(-42*x)/42). - _Vaclav Kotesovec_, Dec 07 2014
%e E.g.f.: A(x) = 1 + 6*x + 539*x^2/2! + 104272*x^3/3! + 31513125*x^4/4! + 13018130762*x^5/5! +...
%e such that A(x) = exp( 7*x*A(x)^6 * G(x*A(x)^6)^6 ) / G(x*A(x)^6),
%e where G(x) = 1 + x*G(x)^7 is the g.f. A002296:
%e G(x) = 1 + x + 7*x^2 + 70*x^3 + 819*x^4 + 10472*x^5 + 141778*x^6 +...
%e Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^6) where
%e F(x) = 1 + 6*x + 107*x^2/2! + 3508*x^3/3! + 171741*x^4/4! + 11280842*x^5/5! +...
%e F(x) = exp( 7*x*G(x)^6 ) / G(x) is the e.g.f. of A251667.
%t Table[(5*n + 1)*(6*n + 1)^(n - 2)*7^n, {n, 0, 50}] (* _G. C. Greubel_, Nov 14 2017 *)
%o (PARI) {a(n) = (5*n+1) * (6*n+1)^(n-2) * 7^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^7 +x*O(x^n));
%o A = ( serreverse( x*G^6 / exp(42*x*G^6) )/x )^(1/6); n!*polcoeff(A, n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (Magma) [(5*n + 1)*(6*n + 1)^(n - 2)*7^n: n in [0..50]]; // _G. C. Greubel_, Nov 14 2017
%Y Cf. A251667, A002296.
%Y Cf. Variants: A127670, A251693, A251694, A251695, A251696, A251698, A251699, A251700.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Dec 07 2014
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