A251697 a(n) = (5*n+1) * (6*n+1)^(n-2) * 7^n.

1, 6, 539, 104272, 31513125, 13018130762, 6835288192159, 4358439870247764, 3271482918202092041, 2826044644022395468750, 2761781119675422226696419, 3012587650584028093856586776, 3628565076873134344787430377389, 4783177086109789054912470697687698, 6849486554475843842876951982177734375

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..248

Formula

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:

(1) A(x) = exp( 7*x*A(x)^6 * G(x*A(x)^6)^6 ) / G(x*A(x)^6).

(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.

(3) A(x) = ( Series_Reversion( x*G(x)^6 / exp(42*x*G(x)^6) )/x )^(1/6).

E.g.f.: (-LambertW(-42*x)/(42*x))^(1/6) * (1 + LambertW(-42*x)/42). - Vaclav Kotesovec, Dec 07 2014

Example

E.g.f.: A(x) = 1 + 6*x + 539*x^2/2! + 104272*x^3/3! + 31513125*x^4/4! + 13018130762*x^5/5! +...
such that A(x) = exp( 7*x*A(x)^6 * G(x*A(x)^6)^6 ) / G(x*A(x)^6),
where G(x) = 1 + x*G(x)^7 is the g.f. A002296:
G(x) = 1 + x + 7*x^2 + 70*x^3 + 819*x^4 + 10472*x^5 + 141778*x^6 +...
Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^6) where
F(x) = 1 + 6*x + 107*x^2/2! + 3508*x^3/3! + 171741*x^4/4! + 11280842*x^5/5! +...
F(x) = exp( 7*x*G(x)^6 ) / G(x) is the e.g.f. of A251667.

Mathematica

Table[(5*n + 1)*(6*n + 1)^(n - 2)*7^n, {n, 0, 50}] (* G. C. Greubel, Nov 14 2017 *)

Prog

(PARI) {a(n) = (5*n+1) * (6*n+1)^(n-2) * 7^n}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=local(G=1, A=1); for(i=0, n, G = 1 + x*G^7 +x*O(x^n));
A = ( serreverse( x*G^6 / exp(42*x*G^6) )/x )^(1/6); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(Magma) [(5*n + 1)*(6*n + 1)^(n - 2)*7^n: n in [0..50]]; // G. C. Greubel, Nov 14 2017

Crossrefs

Cf. A251667, A002296.

Cf. Variants: A127670, A251693, A251694, A251695, A251696, A251698, A251699, A251700.

Sequence in context: A202967 A230330 A252174 * A173789 A258880 A367567

Adjacent sequences: A251694 A251695 A251696 * A251698 A251699 A251700

Keyword

nonn

Author

Paul D. Hanna, Dec 07 2014

Status

approved