cp's OEIS Frontend

A251589 a(n) = 9^(n-7) * (n+1)^(n-9) * (262144*n^7 + 2494464*n^6 + 10470208*n^5 + 25229505*n^4 + 37857568*n^3 + 35537670*n^2 + 19414368*n + 4782969).

%I A251589 #10 Sep 08 2022 08:46:10
%S A251589 1,1,11,312,15261,1082784,101540943,11887094592,1671909996537,
%T A251589 274908280855680,51777320665313331,10995461917161191424,
%U A251589 2599903208658745278549,677539978514149068122112,192961347474468607479984375,59630505640075441195821563904,19874186375795055056211917494257
%N A251589 a(n) = 9^(n-7) * (n+1)^(n-9) * (262144*n^7 + 2494464*n^6 + 10470208*n^5 + 25229505*n^4 + 37857568*n^3 + 35537670*n^2 + 19414368*n + 4782969).
%H A251589 G. C. Greubel, <a href="/A251589/b251589.txt">Table of n, a(n) for n = 0..293</a>
%F A251589 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 A251589 (1) A(x) = exp( 9*x*A(x) * G(x*A(x))^8 ) / G(x*A(x))^8.
%F A251589 (2) A(x) = F(x*A(x)) where F(x) = exp(9*x*G(x)^8)/G(x)^8 is the e.g.f. of A251579.
%F A251589 (3) a(n) = [x^n/n!] F(x)^(n+1)/(n+1) where F(x) is the e.g.f. of A251579.
%F A251589 E.g.f.: -LambertW(-9*x) * (9 + LambertW(-9*x))^8 / (x*9^9). - _Vaclav Kotesovec_, Dec 07 2014
%e A251589 E.g.f.: A(x) = 1 + x + 11*x^2/2! + 312*x^3/3! + 15261*x^4/4! + 1082784*x^5/5! +...
%e A251589 such that A(x) = exp( 9*x*A(x) * G(x*A(x))^8 ) / G(x*A(x))^8
%e A251589 where G(x) = 1 + x*G(x)^9 is the g.f. of A062994:
%e A251589 G(x) = 1 + x + 9*x^2 + 117*x^3 + 1785*x^4 + 29799*x^5 + 527085*x^6 +...
%e A251589 RELATED SERIES.
%e A251589 Note that A(x) = F(x*A(x)) where F(x) = exp(9*x*G(x)^8)/G(x)^8,
%e A251589 F(x) = 1 + x + 9*x^2/2! + 225*x^3/3! + 10017*x^4/4! + 656289*x^5/5! +...
%e A251589 is the e.g.f. of A251579.
%t A251589 Table[9^(n - 7)*(n + 1)^(n - 9)*(262144*n^7 + 2494464*n^6 + 10470208*n^5 + 25229505*n^4 + 37857568*n^3 + 35537670*n^2 + 19414368*n + 4782969), {n, 0, 50}] (* _G. C. Greubel_, Nov 13 2017 *)
%o A251589 (PARI) {a(n) = 9^(n-7) * (n+1)^(n-9) * (262144*n^7 + 2494464*n^6 + 10470208*n^5 + 25229505*n^4 + 37857568*n^3 + 35537670*n^2 + 19414368*n + 4782969)}
%o A251589 for(n=0,20,print1(a(n),", "))
%o A251589 (PARI) {a(n) = local(G=1,A=1); for(i=1,n, G=1+x*G^9 +x*O(x^n));
%o A251589 for(i=1,n, A = exp(9*x*A * subst(G^8,x,x*A) ) / subst(G^8,x,x*A) ); n!*polcoeff(A, n)}
%o A251589 for(n=0, 20, print1(a(n), ", "))
%o A251589 (Magma) [9^(n - 7)*(n + 1)^(n - 9)*(262144*n^7 + 2494464*n^6 + 10470208*n^5 + 25229505*n^4 + 37857568*n^3 + 35537670*n^2 + 19414368*n + 4782969): n in [0..10]]; // _G. C. Greubel_, Nov 13 2017
%Y A251589 Cf. A251579, A062994.
%Y A251589 Cf. Variants: A251583, A251584, A251585, A251586, A251587, A251588, A251590.
%K A251589 nonn
%O A251589 0,3
%A A251589 _Paul D. Hanna_, Dec 06 2014