OFFSET
0,2
COMMENTS
From G. C. Greubel, May 23 2020: (Start)
This sequence is part of a class of sequences, for m >= 0, with the properties:
a(n) = 2*m*(4*m+1)^(n-1) - (1/2)*Sum_{k=1..n-1} a(k)*a(n-k).
a(n) = Sum_{k=0..n} m^k * binomial(n-1, n-k) * binomial(2*k, k).
n*a(n) = 2*((2*m+1)*n - (m+1))*a(n-1) - (4*m+1)*(n-2)*a(n-2).
a(n) = (2*m) * Hypergeometric2F1(-n+1, 3/2; 2; -4*m), for n > 0.
(4*m + 1)^n = Sum_{k=0..n} Sum_{j=0..k} a(j)*a(k-j).
G.f.: sqrt( (1 - t)/(1 - (4*m+1)*t) ).
This sequence is the case of m=3. (End)
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
G.f.: sqrt((1-x)/(1-13*x))
Sum_{i=0..n} Sum_{j=0..i} a(j)*a(i-j) = 13^n.
D-finite with recurrence: n*a(n) = 2*(7*n-4)*a(n-1) - 13*(n-2)*a(n-2). - Vaclav Kotesovec, Oct 14 2012
a(n) ~ 2*sqrt(3)*13^(n-1/2)/sqrt(Pi*n). - Vaclav Kotesovec, Oct 14 2012
a(0) = 1; a(n) = (6/n) * Sum_{k=0..n-1} (n+k) * a(k). - Seiichi Manyama, Mar 28 2023
MAPLE
seq(coeff(series( sqrt((1-x)/(1-13*x)) , x, n+1), x, n), n = 0..30); # G. C. Greubel, May 23 2020
MATHEMATICA
CoefficientList[Series[Sqrt[(1-x)/(1-13x)], {x, 0, 25}], x]
PROG
(PARI) my(x='x+O('x^66)); Vec(sqrt((1-x)/(1-13*x))) \\ Joerg Arndt, May 10 2013
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( Sqrt((1-x)/(1-13*x)) )); // G. C. Greubel, May 23 2020
(Sage)
def A085362_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( sqrt((1-x)/(1-13*x)) ).list()
A085362_list(30) # G. C. Greubel, May 23 2020
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Mario Catalani (mario.catalani(AT)unito.it), Jun 25 2003
STATUS
approved