|
|
A120589
|
|
Self-convolution of A120588, such that a(n) = 3*A120588(n) for n >= 2.
|
|
2
|
|
|
1, 2, 3, 6, 15, 42, 126, 396, 1287, 4290, 14586, 50388, 176358, 624036, 2228700, 8023320, 29084535, 106073010, 388934370, 1432916100, 5301789570, 19692361260, 73398801060, 274447690920, 1029178840950, 3869712441972, 14585839204356
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
For n >= 2, a(n) equals 2^(2n+1) times the coefficient of Pi in 2F1([3/2, n+1], [5/2], -1). - John M. Campbell, Jul 17 2011
|
|
LINKS
|
|
|
FORMULA
|
a(n) = 3*A000108(n-1) for n >= 2, where A000108 are the Catalan numbers.
|
|
EXAMPLE
|
A(x) = 1 + 2*x + 3*x^2 + 6*x^3 + 15*x^4 + 42*x^5 + 126*x^6 + 396*x^7 + ...
A(x)^(1/2) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + ...
|
|
MAPLE
|
A120589List := proc(m) local A, P, n; A := [1, 2, 3]; P := [3];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-1]]);
A := [op(A), P[-1]] od; A end: A120589List(26); # Peter Luschny, Mar 26 2022
|
|
MATHEMATICA
|
Join[{1, 2, 3}, Table[3*(2*n)!/n!/(n+1)!, {n, 2, 40}]]
CoefficientList[Series[(5-2x -3Sqrt[1-4x])/2, {x, 0, 30}], x] (* G. C. Greubel, Feb 18 2019 *)
|
|
PROG
|
(PARI) {a(n)=local(A=1+x+x^2+x*O(x^n)); for(i=0, n, A=A-3*A+2+x+A^2); polcoeff(A^2, n)}
(PARI) my(x='x+O('x^30)); Vec((5-2*x-3*sqrt(1-4*x))/2) \\ G. C. Greubel, Feb 18 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (5-2*x-3*Sqrt(1-4*x))/2 )); // G. C. Greubel, Feb 18 2019
(Sage) ((5-2*x-3*sqrt(1-4*x))/2).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 18 2019
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|