A158119 Unsigned bisection of A157308 and A157310.

1, 1, 3, 38, 947, 37394, 2120190, 162980012, 16330173251, 2070201641498, 324240251016266, 61525045423103316, 13913915097436287598, 3698477457114061621492, 1141824214469896983332508

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

Cf. A157308, A157310, A155585, A157309, A158120.

Sequence in context: A109518 A300631 A300627 * A263332 A335529 A062155

Adjacent sequences: A158116 A158117 A158118 * A158120 A158121 A158122

Keyword

nonn

Author

Paul D. Hanna, Mar 12 2009

Status

approved