OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..200 (terms 0..50 from Paul D. Hanna)
M. Bozejko and T. Hasebe, On free infinite divisibility for classical Meixner distributions, arXiv preprint arXiv:1302.4885 [math.PR], 2013-2014.
FORMULA
G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 2^2*x/(A(x) - 3^2*x/(A(x) - 4^2*x/(A(x) - 5^2*x/(A(x) - 6^2*x/(A(x) - ...)))))), a continued fraction. - Paul D. Hanna, Nov 04 2020
Conjecture: a(m) == 1 (mod 2) iff m is a power of 2 or m=0. [Paul D. Hanna, Mar 16 2009]
a(n) ~ 2^(4*n + 3) * n^(2*n + 1/2) / (Pi^(2*n + 1/2) * exp(2*n)). - Vaclav Kotesovec, Nov 12 2020
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 38*x^3 + 947*x^4 + 37394*x^5 + 2120190*x^6 + 162980012*x^7 + 16330173251*x^8 + 2070201641498*x^9 + 324240251016266*x^10 +...
RELATED FUNCTIONS.
G.f. of A157308, B(x) = x + A(-x^2), satisfies the condition
that both B(x) and F(x) = B(x*F(x)) = o.g.f. of A155585
have zeros for every other coefficient after initial terms:
A157308 = [1,1,-1,0,3,0,-38,0,947,0,-37394,0,2120190,0,...];
A155585 = [1,1,0,-2,0,16,0,-272,0,7936,0,-353792,0,...].
...
G.f. of A157310, C(x) = 2+x - A(-x^2), satisfies the condition
that both C(x) and G(x) = C(x/G(x)) = o.g.f. of A157309
have zeros for every other coefficient after initial terms:
A157310 = [1,1,1,0,-3,0,38,0,-947,0,37394,0,-2120190,0,...];
A157309 = [1,1,0,-1,0,9,0,-176,0,5693,0,-272185,0,...].
...
MATHEMATICA
terms = 30;
F[x_] = Sum[n! x^n/Product[(1 + 2 k x), {k, 1, n}], {n, 0, terms+1}] + O[x]^(terms+1);
A[x_] = x/InverseSeries[x F[x]];
Partition[CoefficientList[A[x], x][[1 ;; terms]], 2][[All, 1]] // Abs (* Jean-François Alcover, Jul 27 2018 *)
PROG
(PARI) {a(n)=local(A=[1, 1]); for(i=1, 2*n, if(#A%2==0, A=concat(A, 0); ); if(#A%2==1, A=concat(A, t); A[ #A]=-subst(Vec(x/serreverse(x*Ser(A)))[ #A], t, 0))); (-1)^n*Vec(x/serreverse(x*Ser(A)))[2*n+1]}
(PARI) {a(n) = my(A=[1], CF=1); for(i=1, n, A=concat(A, 0); for(i=1, #A, CF = Ser(A) - (#A-i+1)^2*x/CF ); A[#A] = -polcoeff(CF, #A-1) ); A[n+1] }
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Nov 04 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 12 2009
STATUS
approved