%I #8 Nov 10 2017 18:15:39

%S 1,1,9,202,7745,429192,31383169,2862173104,313456891041,

%T 40120056928000,5880757402778489,971776774765633536,

%U 178783183332534538849,36248462186643418851328,8031317081954231731640625,1930762515644331053063077888,500587184017640118192794723009,139240118930461640299714951839744

%N a(n) = 7^(n-5) * (n+1)^(n-7) * (1296*n^5 + 9720*n^4 + 30555*n^3 + 50665*n^2 + 44621*n + 16807).

%H G. C. Greubel, <a href="/A251587/b251587.txt">Table of n, a(n) for n = 0..300</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) * G(x*A(x))^6 ) / G(x*A(x))^6.

%F (2) A(x) = F(x*A(x)) where F(x) = exp(7*x*G(x)^6)/G(x)^6 is the e.g.f. of A251577.

%F (3) a(n) = [x^n/n!] F(x)^(n+1)/(n+1) where F(x) is the e.g.f. of A251577.

%F E.g.f.: -LambertW(-7*x) * (7 + LambertW(-7*x))^6 / (x*7^7). - _Vaclav Kotesovec_, Dec 07 2014

%e E.g.f.: A(x) = 1 + x + 9*x^2/2! + 202*x^3/3! + 7745*x^4/4! + 429192*x^5/5! +...

%e such that A(x) = exp( 7*x*A(x) * G(x*A(x))^6 ) / G(x*A(x))^6

%e where G(x) = 1 + x*G(x)^5 is the g.f. of A002296:

%e G(x) = 1 + x + 7*x^2 + 70*x^3 + 819*x^4 + 10472*x^5 + 141778*x^6 +...

%e RELATED SERIES.

%e Note that A(x) = F(x*A(x)) where F(x) = exp(7*x*G(x)^6)/G(x)^6,

%e F(x) = 1 + x + 7*x^2/2! + 133*x^3/3! + 4501*x^4/4! + 224497*x^5/5! +...

%e is the e.g.f. of A251577.

%t Table[7^(n - 5)*(n + 1)^(n - 7)*(1296*n^5 + 9720*n^4 + 30555*n^3 +

%t 50665*n^2 + 44621*n + 16807), {n, 0, 50}] (* _G. C. Greubel_, Nov 10 2017 *)

%o (PARI) {a(n) = 7^(n-5) * (n+1)^(n-7) * (1296*n^5 + 9720*n^4 + 30555*n^3 + 50665*n^2 + 44621*n + 16807)}

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

%o (PARI) {a(n) = local(G=1,A=1); for(i=1,n, G=1+x*G^7 +x*O(x^n));

%o for(i=1,n, A = exp(7*x*A * subst(G^6,x,x*A) ) / subst(G^6,x,x*A) ); n!*polcoeff(A, n)}

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

%Y Cf. A251577, A002296.

%Y Cf. Variants: A251583, A251584, A251585, A251586, A251588, A251589, A251590.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 06 2014