AW: Re: Double factorial related sequence

[Creighton Dement]

> a(n) = 2^n*GAMMA(n+1/2)/Pi^(1/2)*(1/2*Psi(n+1/2)+1/2*gamma+ln(2)+1).
> 
> 
> 
> Rgrds,
> 
> V.

I very much appreciate your observations. With your permission, I will
simply insert your comments  (with reference to you, of course) into the
sequece submission form upon submission. If you'd like to further
cooperate on any sequence submissions (see 2nd note, below), please let
me know. 

Two notes: 
I can also break down the double factorial sequence itself using
floretion techniques:

1lestessumseq: 1, 1, 3, 15, 105, 945, 10395, 135135, 2027025,

1lestessum(pos)seq: 3, 5, 17, 91, 667, 6213, 70233, 933819, 14277555,
246772485, 4757596065, 101218975515, 2355535057995, 59520844736325 

1lestessum(neg)seq:-2, -4, -14, -76, -562, -5268, -59838, -798684,
-12250530, -212313060, -4102866990, -87469664940, -2039300914770,
-51614991155700

We have the identity: 
1lestessum(pos)seq + 1lestessum(neg)seq =  1lestessumseq

Again, it seems both sequences are unlisted. 

2nd note: Here are some additional sequences (along with the e.g.f.
given by Maple) which I would only want to submit with the help of
others:

with(gfun): guessgf([1, -5, 2, 14, -84, 420, -2184, 12264, -74256,
476112, -3118752, 19262880, -83289024, -472949568, 24067225728,
-568467768960, 11696187379968], x);
-5/3*sqrt(3)*sin(sqrt(3)*ln(1+x))+cos(sqrt(3)*ln(1+x)), egf


with(gfun): guessgf([ -1, 1, 2, -10, 36, -132, 504, -1848, 4368, 30576,
-812448, 12868128, -186548544, 2694968640, -39951580032], x);
1/3*sqrt(3)*sin(sqrt(3)*ln(1+x))-cos(sqrt(3)*ln(1+x)), egf


with(gfun): guessgf([ 1, 1, -4, 8, -12, -12, 336, -3360, 30576, -283920,
2778048, -28933632, 321467328, -3805978176, 47893757184, -638503197696,
8985735053568], x);
1/3*sqrt(3)*sin(sqrt(3)*ln(1+x))+cos(sqrt(3)*ln(1+x)), egf

with(gfun): guessgf([ 1, 1, -1, 11, -47, 633, -5073, 86739, -1030815,
21577905, -341061345, 8471746395, -167351545935, 4816256934825,
-114227230079025], x);

(1+1/2*Pi)/sqrt(2*x+1)+I*ln(2*x+sqrt(-1+4*x^2))/sqrt(2*x+1), egf

with(gfun): factor(guessgf([   -1, -1, 3, -9, 33, -147, 777, -4767,
33327, -261513, 2274363, -21689577, 224716401, -2508479883, 29940002001,
-379268237439 ], x));

-(sqrt(3)*sin(1/2*sqrt(3)*ln(1+x))+cos(1/2*sqrt(3)*ln(1+x)))/sqrt(1+x)

with(gfun): factor(guessgf([   -1, 1, -1, 1, 1, -21, 189, -1617, 14511,
-140007, 1460823, -16475823, 200316753, -2616317613, 36574527717 ], x));

1/3*(sqrt(3)*sin(1/2*sqrt(3)*ln(1+x))-3*cos(1/2*sqrt(3)*ln(1+x)))/sqrt(1+x)


Sincerely, 
Creighton 

-It's a shame when the girl of your dreams would still rather be with
someone else when you're actually in a dream.

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