|
| |
|
|
A082366
|
|
G.f.: (1 - 7*x - sqrt(49*x^2 - 18*x + 1))/(2*x).
|
|
6
|
|
|
|
1, 8, 72, 712, 7560, 84616, 985032, 11814728, 145043208, 1813915912, 23029334856, 296050614216, 3846007927944, 50412893051784, 665925356663496, 8855844075949128, 118467982501096968, 1593108078166843912
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
More generally coefficients of (1 - m*x - sqrt(m^2*x^2 - (2*m+4)*x + 1))/(2*x) are given by a(0)=1 and a(n) = (1/n)*Sum_{k=0..n} (m+1)^k*C(n,k)*C(n,k-1) for n > 0.
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(0)=1; a(n) = (1/n)*Sum_{k=0..n} 8^k*C(n, k)*C(n, k-1) for n > 0.
D-finite with recurrence: (n+1)*a(n) + 9*(1-2n)*a(n-1) + 49*(n-2)*a(n-2) = 0. - R. J. Mathar, Dec 08 2011
a(n) ~ sqrt(16+18*sqrt(2))*(9+4*sqrt(2))^n/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012
G.f.: 1/(1 - 7*x - x/(1 - 7*x - x/(1 - 7*x - x/(1 - 7*x - x/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Apr 04 2018
|
|
|
MATHEMATICA
|
CoefficientList[Series[(1-7x-Sqrt[49x^2-18x+1])/(2x), {x, 0, 20}], x] (* Harvey P. Dale, Feb 22 2011 *)
|
|
|
PROG
|
(PARI) a(n)=if(n<1, 1, sum(k=0, n, 8^k*binomial(n, k)*binomial(n, k-1))/n)
(PARI) x='x+O('x^99); Vec((1-7*x-(49*x^2-18*x+1)^(1/2))/(2*x)) \\ Altug Alkan, Apr 04 2018
(GAP) Concatenation([1], List([1..20], n->(1/n)*Sum([0..n], k->8^k*Binomial(n, k)*Binomial(n, k-1)))); # Muniru A Asiru, Apr 05 2018
(Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-7*x-Sqrt(49*x^2-18*x+1))/(2*x))); // G. C. Greubel, Sep 16 2018
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
