# [seqfan] Re: Zeros in A172390 and A172391

Paul D Hanna pauldhanna at juno.com
Mon Mar 29 23:06:44 CEST 2010

```SeqFans,
To expand on Joerg's observation,
here is a relation that both sequences A172390 and A172391 seem to have in common.
No doubt the relation is due to their elliptic connections.

Let
A(x) = g.f. of A172390, and
B(x) = g.f. of A172391;

then the following functions equal the sum of the respective quadrasections defined by:
sqrt( sqrt(A(x^2)) + 4*x ) = QA0(x^4) + x^2*QA1(x^4)
sqrt( sqrt(B(x^2)) + 4*x ) = QB0(x^4) + x^2*QB1(x^4)

such that the quadrasections satisfy the product:
QA0(x^4)*QA1(x^4)  =  QB0(x^4)*QB1(x^4)  =  2
and also
QA0(x^2)^2 + x*QA1(x^2)^2 = sqrt(A(x)) ;
QB0(x^2)^2 + x*QB1(x^2)^2 = sqrt(B(x)).

Is this relation trivial?
Paul
----------------------------------------------------------------
EXAMPLE for A172390.

Convolution square-root of A172390 = A158101, which begins:
[1,4,4,-16,-28,176,336,-2496,-4956,40112,81488,-694720,...].

Interleave A158101 with [4,0,0,0,...] to form A158100:
[1,4,4,0,4,0,-16,0,-28,0,176,0,336,0,-2496,0,-4956,0,40112,0,...].

Convolution square-root of A158100 = A158122, which begins:
[1,2,0,0,2,-4,0,0,-16,40,0,0,200,-544,0,0,-3006,8540,0,0,...]

in which the convolution of the non-zero quadrasections equals 2:
[1,2,-16,200,-3006,...]*[2,-4,40,-544,8540,...] = 2.

Note also that the quadrasections are square-roots of the bisections of A172390^(1/2):
[1,2,-16,200,-3006,...]^2 = [1, 4, -28, 336, -4956,...];
[2,-4,40,-544,8540,...]^2 = [4, -16, 176, -2496, 40112,...].
----------------------------------------------------------------
EXAMPLE for A172391.

Convolution square-root of A172391 = A172393, which begins:
[1,4,-2,8,-20,96,-324,1648,-6348,33200,-137848,...].

Interleave A172393 with [4,0,0,0,...] to form:
[1,4,4,0,-2,0,8,0,-20,0,96,0,-324,0,1648,0,-6348,0,33200,0,-137848,0,...]

Convolution square-root of this result begins:
[1,2,0,0,-1,2,0,0,-21/2,23,0,0,-345/2,389,0,0,-27213/8,31115/4,0,0,-593095/8,...]

in which the convolution of the non-zero quadrasections equals 2:
[1,-1,-21/2,-345/2,-27213/8,-593095/8...]*[2,2,23,389,31115/4,683631/4,...] = 2.

Note also that the quadrasections are sqaure-roots of the bisections of A172391^(1/2):
[1,-1,-21/2,-345/2,-27213/8,-593095/8...]^2 = [1, -2, -20, -324, -6348, -137848,...];
[2, 2, 23, 389, 31115/4, 683631/4,...]^2 = [4, 8, 96, 1648, 33200, 732640,...].
----------------------------------------------------------------
[END]

```