A251590 a(n) = 10^(n-8) * (n+1)^(n-10) * (4782969*n^8 + 50309748*n^7 + 237013938*n^6 + 655232760*n^5 + 1166624361*n^4 + 1374998212*n^3 + 1051760172*n^2 + 479277840*n + 100000000).

1, 1, 12, 376, 20384, 1604880, 167097280, 21724557760, 3393929393280, 619917062849920, 129708290461760000, 30601444681382400000, 8038933665661600000000, 2327544788978773504000000, 736481767572932400000000000, 252867957872989831168000000000, 93638020040848371127040000000000

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..289

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) * G(x*A(x))^9 ) / G(x*A(x))^9.

(2) A(x) = F(x*A(x)) where F(x) = exp(10*x*G(x)^9)/G(x)^9 is the e.g.f. of A251580.

(3) a(n) = [x^n/n!] F(x)^(n+1)/(n+1) where F(x) is the e.g.f. of A251580.

E.g.f.: -LambertW(-10*x) * (10 + LambertW(-10*x))^9 / (x*10^10). - Vaclav Kotesovec, Dec 07 2014

Example

E.g.f.: A(x) = 1 + x + 12*x^2/2! + 376*x^3/3! + 20384*x^4/4! + 1604880*x^5/5! +...
such that A(x) = exp( 10*x*A(x) * G(x*A(x))^9 ) / G(x*A(x))^9
where G(x) = 1 + x*G(x)^10 is the g.f. of A059968:
G(x) = 1 + x + 10*x^2 + 145*x^3 + 2470*x^4 + 46060*x^5 + 910252*x^6 +...
RELATED SERIES.
Note that A(x) = F(x*A(x)) where F(x) = exp(9*x*G(x)^8)/G(x)^8,
F(x) = 1 + x + 10*x^2/2! + 280*x^3/3! + 13960*x^4/4! + 1023760*x^5/5! +...
is the e.g.f. of A251580.

Mathematica

Table[10^(n - 8)*(n + 1)^(n - 10)*(4782969*n^8 + 50309748*n^7 + 237013938*n^6 + 655232760*n^5 + 1166624361*n^4 + 1374998212*n^3 + 1051760172*n^2 + 479277840*n + 100000000), {n, 0, 50}] (* G. C. Greubel, Nov 13 2017 *)

Prog

(PARI) {a(n) = 10^(n-8) * (n+1)^(n-10) * (4782969*n^8 + 50309748*n^7 + 237013938*n^6 + 655232760*n^5 + 1166624361*n^4 + 1374998212*n^3 + 1051760172*n^2 + 479277840*n + 100000000)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = local(G=1, A=1); for(i=1, n, G=1+x*G^10 +x*O(x^n));
for(i=1, n, A = exp(10*x*A * subst(G^9, x, x*A) ) / subst(G^9, x, x*A) ); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(Magma) [10^(n - 8)*(n + 1)^(n - 10)*(4782969*n^8 + 50309748*n^7 + 237013938*n^6 + 655232760*n^5 + 1166624361*n^4 + 1374998212*n^3 + 1051760172*n^2 + 479277840*n + 100000000): n in [0..50]]; // G. C. Greubel, Nov 13 2017

Crossrefs

Cf. A251580, A059968.

Cf. Variants: A251583, A251584, A251585, A251586, A251587, A251588, A251589.

Sequence in context: A177112 A081021 A269609 * A352651 A196459 A193132

Adjacent sequences: A251587 A251588 A251589 * A251591 A251592 A251593

Keyword

nonn

Author

Paul D. Hanna, Dec 06 2014

Status

approved