login
a(n) = (7*n+1) * (8*n+1)^(n-2) * 9^n.
9

%I #12 Sep 08 2022 08:46:10

%S 1,8,1215,400950,207202941,146509780644,131737019154363,

%T 143893722808861650,185052548743241765625,273919266332571877019712,

%U 458736814135093804224189111,857575304752878031562956215918,1770298011965146072399475770453365,3999656915702652258291935606835937500

%N a(n) = (7*n+1) * (8*n+1)^(n-2) * 9^n.

%H G. C. Greubel, <a href="/A251699/b251699.txt">Table of n, a(n) for n = 0..235</a>

%F 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:

%F (1) A(x) = exp( 9*x*A(x)^8 * G(x*A(x)^8)^8 ) / G(x*A(x)^8).

%F (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.

%F (3) A(x) = ( Series_Reversion( x*G(x)^8 / exp(72*x*G(x)^8) )/x )^(1/8).

%F E.g.f.: (-LambertW(-72*x)/(72*x))^(1/8) * (1 + LambertW(-72*x)/72). - _Vaclav Kotesovec_, Dec 07 2014

%e E.g.f.: A(x) = 1 + 8*x + 1215*x^2/2! + 400950*x^3/3! + 207202941*x^4/4! + 146509780644*x^5/5! +...

%e such that A(x) = exp( 9*x*A(x)^8 * G(x*A(x)^8)^8 ) / G(x*A(x)^8),

%e where G(x) = 1 + x*G(x)^9 is the g.f. A062994:

%e G(x) = 1 + x + 9*x^2 + 117*x^3 + 1785*x^4 + 29799*x^5 + 527085*x^6 +...

%e Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^8) where

%e F(x) = 1 + 8*x + 191*x^2/2! + 8310*x^3/3! + 537117*x^4/4! + 46444164*x^5/5! +...

%e F(x) = exp( 9*x*G(x)^8 ) / G(x) is the e.g.f. of A251669.

%t Table[(7*n + 1)*(8*n + 1)^(n - 2)*9^n, {n, 0, 50}] (* _G. C. Greubel_, Nov 14 2017 *)

%o (PARI) {a(n) = (7*n+1) * (8*n+1)^(n-2) * 9^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^9 +x*O(x^n));

%o A = ( serreverse( x*G^8 / exp(72*x*G^8) )/x )^(1/8); n!*polcoeff(A, n)}

%o for(n=0, 20, print1(a(n), ", "))

%o (Magma) [(7*n + 1)*(8*n + 1)^(n - 2)*9^n: n in [0..50]]; // _G. C. Greubel_, Nov 14 2017

%Y Cf. A251669, A062994.

%Y Cf. Variants: A127670, A251693, A251694, A251695, A251696, A251697, A251698, A251700.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 07 2014