E.G.F. for Squared Terms of Special Sequences

[Paul D. Hanna]

Seqfans, 
      Here is a simple formula for sequences formed from the squared
terms of a special family of sequences. 
It was inspired by Michael Somos's observation that
A111883(n)=A000085(n)^2. 
  
There are 7 new sequences below along with a few others in OEIS that
serve as examples. 
 
Would anyone like to prove the conjecture below?  
     Paul 
---------------------------------------------------------------
Conjecture.
The term-by-term square of the sequence generated by: 
     E.g.f.: exp(x + m/2*x^2) 
yields the sequence given by: 
     E.g.f.: exp( x/(1-m*x) )/sqrt(1 - m^2*x^2) 
and is an integer sequence whenever m is an integer. 
---------------------------------------------------------------
EXAMPLES: 

m=1:
SQUARED TERMS OF SEQUENCE:
E.g.f.: exp(x+1/2*x^2) :
http://www.research.att.com/projects/OEIS?Anum=A000085
1,1,2,4,10,26,76,232,764,2620,9496,35696,140152,568504,

IS GIVEN BY:
E.g.f.: exp(x/(1-x))/sqrt(1-x^2) :
http://www.research.att.com/projects/OEIS?Anum=A111883
1,1,4,16,100,676,5776,53824,583696,6864400,90174016,
---------------------------------------------------------
m=-1: 
 SQUARED TERMS  OF SEQUENCE: 
E.g.f.: exp(-x-1/2*x^2) :
http://www.research.att.com/projects/OEIS?Anum=A001464
1,-1,0,2,-2,-6,16,20,-132,-28,1216,-936,-12440,23672,138048,

IS GIVEN BY:
E.g.f.: exp(x/(1+x))/sqrt(1-x^2) :
http://www.research.att.com/projects/OEIS?Anum=A111882
1,1,0,4,4,36,256,400,17424,784,1478656,876096,154753600,
---------------------------------------------------------
m=2:
SQUARED TERMS OF SEQUENCE: 
E.g.f.: exp(x+x^2) :
http://www.research.att.com/projects/OEIS?Anum=A047974
1,1,3,7,25,81,331,1303,5937,26785,133651,669351,3609673,

IS GIVEN BY:
E.g.f.: exp(x/(1-2*x))/sqrt(1-4*x^2) :
(not in OEIS)
1,1,9,49,625,6561,109561,1697809,35247969,717436225,
17862589801,448030761201,13029739166929,387070092765409,
12888060720104025,441427773256896721,16566268858818121921,
641658452161285040769,26803156413926425274569,
(PARI) 
a(n)=n!*polcoeff(exp(x/(1-2*x+x*O(x^n)))/sqrt(1-4*x^2+x*O(x^n)),n)
---------------------------------------------------------
m=3:
SQUARED TERMS OF SEQUENCE: 
E.g.f.: exp(x+3/2*x^2) :
(not in OEIS)
1,1,4,10,46,166,856,3844,21820,114076,703216,4125496,
27331624,175849480,1241782816,8627460976,64507687696,
478625814544,3768517887040,29614311872416,244419831433696,
2021278543778656,17419727924101504,150824111813492800,
(PARI) 
a(n)=n!*polcoeff(exp(x+3/2*x^2+x*O(x^n)),n)
  
IS GIVEN BY:
E.g.f: exp(x/(1-3*x))/sqrt(1-9*x^2) :
(not in OEIS)
1,1,16,100,2116,27556,732736,14776336,476112400,13013333776,
494512742656,17019717246016,747017670477376,
30923039616270400,1542024562112889856,74433082892402872576,
4161241771884669788416,229082670347907481927936,
(PARI) 
a(n)=n!*polcoeff(exp(x/(1-3*x+x*O(x^n)))/sqrt(1-9*x^2+x*O(x^n)),n)
---------------------------------------------------------
m=4:
SQUARED TERMS OF SEQUENCE: 
E.g.f.: exp(x+2*x^2).
(not in OEIS)
1,1,5,13,73,281,1741,8485,57233,328753,2389141,15539261,
120661465,866545993,7140942173,55667517781,484124048161,
4046845186145,36967280461093,328340133863533,
3137853448906601,29405064157989241,292984753866143725,
(PARI) 
a(n)=n!*polcoeff(exp(x+2*x^2+x*O(x^n)),n)
 
IS GIVEN BY: 
E.g.f: exp(x/(1-4*x))/sqrt(1-16*x^2).
(not in OEIS)
1,1,25,169,5329,78961,3031081,71995225,3275616289,
108078535009,5707994717881,241468632426121,
14559189135946225,750901957984356049,50993055118129961929,
3098872535897951163961,234376094007794247481921,
(PARI) 
a(n)=n!*polcoeff(exp(x/(1-4*x+x*O(x^n)))/sqrt(1-16*x^2+x*O(x^n)),n)
---------------------------------------------------------
m=5:
SQUARED TERMS OF SEQUENCE: 
E.g.f.: exp(x+5/2*x^2).
(not in OEIS)
1,1,6,16,106,426,3076,15856,123516,757756,6315976,44203776,
391582456,3043809016,28496668656,241563299776,2378813448976,
21703877431056,223903020594016,2177251989389056,
23448038945820576,241173237884726176,2703217327195886656,
(PARI) 
a(n)=n!*polcoeff(exp(x+5/2*x^2+x*O(x^n)),n)
 
IS GIVEN BY: 
E.g.f: exp(x/(1-5*x))/sqrt(1-25*x^2).
(not in OEIS)
1,1,36,256,11236,181476,9461776,251412736,15256202256,
574194155536,39891552832576,1953973812658176,
153336819846991936,9264773325882888256,
812060124489852846336,58352827798669641650176,
(PARI) 
a(n)=n!*polcoeff(exp(x/(1-5*x+x*O(x^n)))/sqrt(1-25*x^2+x*O(x^n)),n)
---------------------------------------------------------
END
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