|
|
A085363
|
|
a(0)=1, for n>0: a(n) = 4*9^(n-1) - (1/2)*Sum_{i=1..n-1} a(i)*a(n-i).
|
|
6
|
|
|
1, 4, 28, 212, 1676, 13604, 112380, 940020, 7936620, 67494980, 577309148, 4961187092, 42801458764, 370478720356, 3215827927228, 27982214082612, 244004165618220, 2131710838837380, 18654504783815580, 163488269572628820
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
Apparently, the number of 2-D directed walks of semilength n starting at (0,0) and ending on the X-axis using steps NE, SE, NW and SW avoiding adjacent NW/SE and adjacent NE/SW. - David Scambler, Jun 20 2013
Form an array with m(0,n) = m(n,0) = 2^n; m(i,j) equals the sum of the terms to the left of m(i,j) and the sum of the terms above m(i,j), which is m(i,j) = Sum_{k-0..j-1} m(i,k) + Sum_{k=0..i-1} m(k,j). m(n,n) = a(n). - J. M. Bergot, Jul 10 2013
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).
a(n) = (2*m) * Hypergeometric2F1(-n+1, 3/2; 2; -4*m), for n > 0.
n*a(n) = 2*((2*m+1)*n - (m+1))*a(n-1) - (4*m+1)*(n-2)*a(n-2).
(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=2. (End)
The number of elements in the free group on two generators of length 2n that are zero exponent sum. - Tey Berendschot, Aug 09 2021
|
|
LINKS
|
|
|
FORMULA
|
G.f.: sqrt((1-x)/(1-9*x)).
Sum_{i=0..n} Sum_{j=0..i} a(j)*a(i-j) = 9^n.
a(n) = Sum_{k=1..n} 2^k * binomial(n-1, k-1) * binomial(2*k, k). (End)
D-finite with recurrence n*a(n) = (10*n-6)*a(n-1) - (9*n-18)*a(n-2). - Vladeta Jovovic, Jul 16 2004
a(0) = 1; a(n) = (4/n) * Sum_{k=0..n-1} (n+k) * a(k). - Seiichi Manyama, Mar 28 2023
|
|
MAPLE
|
seq(coeff(series(sqrt((1-x)/(1-9*x)), x, n+1), x, n), n = 0..30); # G. C. Greubel, May 23 2020
|
|
MATHEMATICA
|
CoefficientList[Series[Sqrt[(1-x)/(1-9x)], {x, 0, 25}], x]
|
|
PROG
|
(PARI) my(x='x+O('x^66)); Vec(sqrt((1-x)/(1-9*x)) ) \\ Joerg Arndt, May 10 2013
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( Sqrt((1-x)/(1-9*x)) )); // G. C. Greubel, May 23 2020
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( sqrt((1-x)/(1-9*x)) ).list()
|
|
CROSSREFS
|
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
Mario Catalani (mario.catalani(AT)unito.it), Jun 25 2003
|
|
STATUS
|
approved
|
|
|
|