A251588 a(n) = 8^(n-6) * (n+1)^(n-8) * (16807*n^6 + 143031*n^5 + 525875*n^4 + 1074745*n^3 + 1294846*n^2 + 876856*n + 262144).

1, 1, 10, 254, 11080, 700008, 58411696, 6082359760, 760774053888, 111229735731200, 18626295180427264, 3516652429787529216, 739238816214490808320, 171262175332556483854336, 43359709355122360320000000, 11911510903698787868252045312, 3529104034183977458725537447936, 1121766516051874786454563454976000

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..296

Formula

Let G(x) = 1 + x*G(x)^8 be the g.f. of A007556, then the e.g.f. A(x) of this sequence satisfies:

(1) A(x) = exp( 8*x*A(x) * G(x*A(x))^7 ) / G(x*A(x))^7.

(2) A(x) = F(x*A(x)) where F(x) = exp(8*x*G(x)^7)/G(x)^7 is the e.g.f. of A251578.

(3) a(n) = [x^n/n!] F(x)^(n+1)/(n+1) where F(x) is the e.g.f. of A251578.

E.g.f.: -LambertW(-8*x) * (8 + LambertW(-8*x))^7 / (x*8^8). - Vaclav Kotesovec, Dec 07 2014

Example

E.g.f.: A(x) = 1 + x + 10*x^2/2! + 254*x^3/3! + 11080*x^4/4! + 700008*x^5/5! +...
such that A(x) = exp( 7*x*A(x) * G(x*A(x))^6 ) / G(x*A(x))^6
where G(x) = 1 + x*G(x)^8 is the g.f. of A007556:
G(x) = 1 + x + 8*x^2 + 92*x^3 + 1240*x^4 + 18278*x^5 + 285384*x^6 +...
RELATED SERIES.
Note that A(x) = F(x*A(x)) where F(x) = exp(8*x*G(x)^7)/G(x)^7,
F(x) = 1 + x + 8*x^2/2! + 176*x^3/3! + 6896*x^4/4! + 397888*x^5/5! +...
is the e.g.f. of A251578.

Mathematica

Table[8^(n - 6)*(n + 1)^(n - 8)*(16807*n^6 + 143031*n^5 + 525875*n^4 + 1074745*n^3 + 1294846*n^2 + 876856*n + 262144), {n, 0, 50}] (* G. C. Greubel, Nov 13 2017 *)

Prog

(PARI) {a(n) = 8^(n-6) * (n+1)^(n-8) * (16807*n^6 + 143031*n^5 + 525875*n^4 + 1074745*n^3 + 1294846*n^2 + 876856*n + 262144)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = local(G=1, A=1); for(i=1, n, G=1+x*G^8 +x*O(x^n));
for(i=1, n, A = exp(8*x*A * subst(G^7, x, x*A) ) / subst(G^7, x, x*A) ); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(Magma) [8^(n - 6)*(n + 1)^(n - 8)*(16807*n^6 + 143031*n^5 + 525875*n^4 + 1074745*n^3 + 1294846*n^2 + 876856*n + 262144): n in [0..50]]; // G. C. Greubel, Nov 13 2017

Crossrefs

Cf. A251578, A007556.

Cf. Variants: A251583, A251584, A251585, A251586, A251587, A251589, A251590.

Sequence in context: A095983 A059166 A100743 * A126468 A336665 A024293

Adjacent sequences: A251585 A251586 A251587 * A251589 A251590 A251591

Keyword

nonn

Author

Paul D. Hanna, Dec 06 2014

Status

approved