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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).
9
1, 1, 12, 376, 20384, 1604880, 167097280, 21724557760, 3393929393280, 619917062849920, 129708290461760000, 30601444681382400000, 8038933665661600000000, 2327544788978773504000000, 736481767572932400000000000, 252867957872989831168000000000, 93638020040848371127040000000000
OFFSET
0,3
LINKS
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 06 2014
STATUS
approved