|
| |
|
|
A367389
|
|
Expansion of g.f. A(x) = B(x^2)/(1 - 2*x*B(x^2)) where B(x) = 1 + 2*x*B(x)^3 is the g.f. of A153231.
|
|
0
|
|
|
|
1, 2, 6, 16, 52, 152, 512, 1568, 5392, 16992, 59232, 190336, 669952, 2183680, 7742464, 25512448, 90974464, 302368256, 1083175424, 3625435136, 13036688384, 43889186816, 158323564544, 535639556096, 1937483350016, 6582584115200, 23865932414976, 81381420826624, 295661476642816
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = B(x^2)/(1 - 2*x*B(x^2)) where B(x) = 1 + 2*x*B(x)^3 is the g.f. of A153231.
(2) A(x) = 1/(1-2*x - Sum_{n>=1} 2^n * binomial(3*n-1,n)/(3*n-1) * x^(2*n) ).
(3) A(x) = 1/(-2*x + x/Series_Reversion( x - 2*x^3 )).
(4) [x^(n-1)] (1 + (n+1)*x*A(x))^n / A(x)^n = n*(n-1)^(n-2) for n > 1.
(5) [x^(n-1)] (1 + n*x*A(x))^n / A(x)^n = -n*(n-2)^(n-2) for n > 1.
(6) [x^(n-1)] (1 + (n-2)*x*A(x))^n / A(x)^n = -n*(5*n-14)*(n-4)^(n-3) for n >= 1.
a(n) ~ (15*sqrt(3/2)/2 + 9 + (15*sqrt(3/2)/2 - 9)*(-1)^n) * 3^(3*n/2) / (sqrt(Pi) * n^(3/2) * 2^(n/2)). - Vaclav Kotesovec, Dec 24 2023
|
|
|
EXAMPLE
|
G.f.: A(x) = 1 + 2*x + 6*x^2 + 16*x^3 + 52*x^4 + 152*x^5 + 512*x^6 + 1568*x^7 + 5392*x^8 + 16992*x^9 + 59232*x^10 + ...
where
1/A(x) = 1 - 2*x - 2*x^2 - 8*x^4 - 56*x^6 - 480*x^8 - 4576*x^10 - 46592*x^12 - ... - 2^n*binomial(3*n-1,n)/(3*n-1) * x^(2*n) - ...
|
|
|
MATHEMATICA
|
CoefficientList[1/(-2*x + x/InverseSeries[Series[x - 2*x^3, {x, 0, 30}], x]), x] (* Vaclav Kotesovec, Dec 24 2023 *)
|
|
|
PROG
|
(PARI) {a(n) = my(A = 1/(-2*x + x/serreverse(x - 2*x^3 + O(x^(n+2))))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A;
A[#A] = polcoeff( (1 + (m+1)*x*Ser(A))^m / Ser(A)^m , m-1)/m - (m-1)^(m-2) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
