A251700 a(n) = (8*n+1) * (9*n+1)^(n-2) * 10^n.

1, 9, 1700, 700000, 451770000, 399077600000, 448380625000000, 612032839680000000, 983672470878500000000, 1819783935144064000000000, 3809045473682732010000000000, 8900000000000000000000000000000, 22963427643543537169897000000000000, 64847222511564256853913600000000000000

Offset

0,2

Comments

In general, sequence a(n) = (b*n+1)*(c*n+1)^(n-2)*d^n, b > 0, c > 0, d > 0, has e.g.f. (1 + (1-b/c)*LambertW(-c*d*x)/(c+1)) * (-LambertW(-c*d*x)/(c*d*x))^(1/c). - Vaclav Kotesovec, Dec 07 2014

Links

G. C. Greubel, Table of n, a(n) for n = 0..230

Formula

Let G(x) = 1 + x*G(x)^10 be the g.f. of A059968, then the e.g.f. A(x) of this sequence satisfies:

(1) A(x) = exp( 10*x*A(x)^9 * G(x*A(x)^9)^9 ) / G(x*A(x)^9).

(2) A(x) = F(x*A(x)^9) where F(x) = exp(10*x*G(x)^9)/G(x) is the e.g.f. of A251670.

(3) A(x) = ( Series_Reversion( x*G(x)^9 / exp(90*x*G(x)^9) )/x )^(1/9).

E.g.f.: (-LambertW(-90*x)/(90*x))^(1/9) * (1 + LambertW(-90*x)/90). - Vaclav Kotesovec, Dec 07 2014

Example

E.g.f.: A(x) = 1 + 9*x + 1700*x^2/2! + 700000*x^3/3! + 451770000*x^4/4! + 399077600000*x^5/5! +...
such that A(x) = exp( 10*x*A(x)^9 * G(x*A(x)^9)^9 ) / G(x*A(x)^9),
where G(x) = 1 + x*G(x)^10 is the g.f. A059968:
G(x) = 1 + x + 10*x^2 + 145*x^3 + 2470*x^4 + 46060*x^5 + 910252*x^6 +...
Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^9) where
F(x) = 1 + 9*x + 242*x^2/2! + 11824*x^3/3! + 856824*x^4/4! + 82986080*x^5/5! +...
F(x) = exp( 10*x*G(x)^9 ) / G(x) is the e.g.f. of A251670.

Mathematica

Table[(8*n + 1)*(9*n + 1)^(n - 2)*10^n, {n, 0, 50}] (* G. C. Greubel, Nov 14 2017 *)

Prog

(PARI) {a(n) = (8*n+1) * (9*n+1)^(n-2) * 10^n}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=local(G=1, A=1); for(i=0, n, G = 1 + x*G^10 +x*O(x^n));
A = ( serreverse( x*G^9 / exp(90*x*G^9) )/x )^(1/9); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(Magma) [(8*n + 1)*(9*n + 1)^(n - 2)*10^n: n in [0..50]]; // G. C. Greubel, Nov 14 2017

Crossrefs

Cf. A251670, A059968.

Cf. Variants: A127670, A251693, A251694, A251695, A251696, A251697, A251698, A251699.

Sequence in context: A197804 A047944 A322245 * A246115 A068182 A368368

Adjacent sequences: A251697 A251698 A251699 * A251701 A251702 A251703

Keyword

nonn

Author

Paul D. Hanna, Dec 07 2014

Status

approved