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<DIV>Seqfans, </DIV>
<DIV> There is much more to this
study that needs to be explored. </DIV>
<DIV>Here is my most general formula so far: </DIV>
<DIV> <BR>(9) Let F,G,H, be formal power series in x such that F(0)=1,
G(0)=1, <BR>then <BR>Sum_{n>=0} m^n * H(q^n*x) * log( F(q^n*x)*G(x) )^n / n!
= <BR>Sum_{n>=0} x^n * G(x)^(m*q^n) * [y^n] H(y)*F(y)^(m*q^n). </DIV>
<DIV> </DIV>
<DIV>But there are yet many undiscovered formulas for sums of
this type, </DIV>
<DIV>such as these examples: <BR>(10) Sum_{n>=0} log((1 + a*p^n*x)*(1 +
b*q^n*x))^n/n! = ? </DIV>
<DIV>(11) Sum_{n>=0} log(1 + (a*p^n + b*q^n)*x)^n/n! = ? </DIV>
<DIV>both yield series with integer coefficients whenever p,q,a,b, are
integer. </DIV>
<DIV>These lead to much more general sums that seem impossible to find a formula
for. <BR>Yet it intrigues me that these sums yield integer series for
integer arguments. <BR> </DIV>
<DIV>Further, not just an exponential sum of powers of logs can be studied;
</DIV>
<DIV>it appears that other functions with their inverses also yield integer
series. </DIV>
<DIV> </DIV>
<DIV> </DIV>
<DIV>Here is an example using the hyperbolic sine series applied on the
inverse sinh: </DIV>
<DIV><BR>(12) G.f.: A(x) = Sum_{n>=0} asinh( 2^(2n+1)*x )^(2n+1) /
(2n+1)! = <BR>2*x + 84*x^3 + 276892*x^5 + 111457917800*x^7 +
<BR>6660816097416169260*x^9 + 66597307693046550483175282456*x^11
+<BR>120167520447600665027319450022840022638104*x^13 +...</DIV>
<DIV> </DIV>
<DIV>Is there a simple formula for the integer coefficients on the right side of
(12)? <BR> </DIV>
<DIV> </DIV>
<DIV>Of course (12) is an example of the more general: </DIV>
<DIV>(13) G.f.: A(x) = Sum_{n>=0} asinh( q^(2n+1)*x )^(2n+1) / (2n+1)!
= ? <BR>yielding some unknown integer series for all integer q. </DIV>
<DIV> </DIV>
<DIV>Is there a simple formula for the integer coefficients on the right side of
(13)? </DIV>
<DIV>It would be nice if it turned out to be as simple a formula as:
</DIV>
<DIV>(2) Sum_{n>=0} log(1 + q^n*x)^n/n! = Sum_{n>=0}
C(q^n,n)*x^n. </DIV>
<DIV> </DIV>
<DIV> </DIV>
<DIV>What other functions can be used besides exp(), sinh(), etc., </DIV>
<DIV>in a similar nontrivial sum that yields an integer series
for all integer q? </DIV>
<DIV> </DIV>
<DIV>Unless I find a breakthrough, this will be the last I post regarding these
sums. </DIV>
<DIV> <BR>But I would like very much to hear from anyone with
answers to the above </DIV>
<DIV>questions or a nice application of any of these sums. </DIV>
<DIV> </DIV>
<DIV>Thanks, </DIV>
<DIV> Paul </DIV>
<DIV> </DIV>
<DIV> </DIV>
<DIV>On Tue, 8 Jan 2008 22:50:04 -0500 <A
href="mailto:pauldhanna@juno.com">pauldhanna@juno.com</A> writes:</DIV>
<BLOCKQUOTE dir=ltr
style="PADDING-LEFT: 10px; MARGIN-LEFT: 10px; BORDER-LEFT: #000000 2px solid">
<DIV>SeqFans, </DIV>
<DIV> A little more on the generalization.
<BR>Introduce a function G(x) (independent of summation index n):
</DIV>
<DIV> </DIV>
<DIV>
<DIV>Let F(x) be any formal power series in x such that F(0)=1; then </DIV>
<DIV>
<DIV>(7) Sum_{n>=0} m^n * F(q^n*x)^b * log( G(x)*F(q^n*x) )^n / n!
= </DIV>
<DIV> Sum_{n>=0} x^n * G(x)^(m*q^n) * [y^n]
F(y)^(m*q^n + b) </DIV>
<DIV> </DIV>
<DIV>where [y^n] F(y) denotes the coefficient of y^n in F(y).</DIV>
<DIV> </DIV>
<DIV>It seems counter-intuitive that the power of G would be m*q^n;
</DIV>
<DIV>however, it can be deduced from (5) (in prior email below).
</DIV></DIV></DIV>
<DIV> </DIV>
<DIV>EXAMPLE of (7). <BR>When applying (7) to A136554:
</DIV>
<DIV>1,3,10,82,2304,232088,81639942,99425060368,421915147527984,<BR>G.f.: A(x)
= Sum_{n>=0} log( (1 + x)*(1 + 2^n*x) )^n / n!.<BR> </DIV>
<DIV>it reveals the nice formulas:</DIV>
<DIV><BR>G.f.: A(x) = Sum_{n>=0} C(2^n,n) * x^n * (1+x)^(2^n). </DIV>
<DIV> </DIV>
<DIV>a(n) = Sum_{k=0..n} C(2^k,k) * C(2^k,n-k). <BR> </DIV>
<DIV>So, from (7) we find a nice simplification for: </DIV>
<DIV>G.f.: A(x) = Sum_{n>=0} log( z*(1 + 2^n*x) )^n / n!<BR>as:
</DIV>
<DIV>G.f.: A(x) = Sum_{n>=0} C(2^n,n) * x^n * z^(2^n)</DIV>
<DIV> </DIV>
<DIV> </DIV>
<DIV>FURTHER YET ... </DIV>
<DIV>But here is where I get stumped. </DIV>
<DIV>Can the following sum be simplified into a form similar to (7)?
</DIV>
<DIV> </DIV>
<DIV>(8) Sum_{n>=0} log( F(p^n*x)*G(q^n*x) )^n / n! = ? </DIV>
<DIV> </DIV>
<DIV>This sum again returns an integer series in x when both </DIV>
<DIV>F(x) and G(x) are integer series in x for all integer p, q.
</DIV>
<DIV> </DIV>
<DIV>But simplifying (8) into a form like (7) when p not= q </DIV>
<DIV>is not straitforward (at least for me). </DIV>
<DIV> </DIV>
<DIV>Perhaps someone can find a nice form for (8)? </DIV>
<DIV> </DIV>
<DIV>EXAMPLE of (8).</DIV>
<DIV>Let p=2, q=3, then we have A136578:</DIV>
<DIV>G.f.: A(x) = Sum_{n>=0} log( (1 + 2^n*x)*(1 + 3^n*x) )^n / n!.</DIV>
<DIV>1,5,78,6527,3450452,12594729052,338284182093366,70004091118158663618,</DIV>
<DIV> </DIV>
<DIV>But I have not found a nice formula for even this simple
case ... </DIV>
<DIV> Paul </DIV>
<DIV> </DIV>
<DIV>On Wed, 2 Jan 2008 21:50:02 -0500 <A
href="mailto:pauldhanna@juno.com">pauldhanna@juno.com</A> writes:</DIV>
<BLOCKQUOTE dir=ltr
style="PADDING-LEFT: 10px; MARGIN-LEFT: 10px; BORDER-LEFT: #000000 2px solid">
<DIV>Seqfans,</DIV>
<DIV>
<DIV> The identity can be generalized further.
</DIV>
<DIV>Let F(x) be any formal power series in x such that F(0)=1. </DIV>
<DIV>Then </DIV>
<DIV>
<DIV>(5) Sum_{n>=0} m^n * F(q^n*x)^b * log( F(q^n*x) )^n / n!
= </DIV>
<DIV> Sum_{n>=0} x^n * [y^n]
F(y)^(m*q^n + b) </DIV>
<DIV> </DIV>
<DIV>where [y^n] G(y) denotes the coefficient of y^n in G(y).</DIV></DIV>
<DIV> </DIV>
<DIV>If we let F(x) = exp(x), then we have the nontrivial result </DIV>
<DIV> </DIV>
<DIV>
<DIV>(6) Sum_{n>=0} m^n * q^(n^2) * exp(b*q^n*x) * x^n / n! =
</DIV>
<DIV> Sum_{n>=0} (m*q^n + b)^n * x^n /
n! </DIV>
<DIV> </DIV>
<DIV>which becomes trivial at b=0. </DIV></DIV>
<DIV></DIV>
<DIV>Example of (6): q=2, m=1, b=1:</DIV>
<DIV>exp(x) + 2*exp(2x) + 2^4*exp(4x)*x^2/2! + 2^9*exp(8x)*x^3/3!
+...+</DIV>
<DIV> 2^(n^2)*exp(2^n*x)/n! +...</DIV>
<DIV>
<DIV>= 1 + 3x + 5^2*x^2/2! + 9^3*x^3/3! + 17^4*x^4/4! +...+ </DIV>
<DIV> (2^n+1)^n*x^n/n! +... </DIV></DIV>
<DIV> </DIV>
<DIV>
<DIV>Example of (6): q=2, m=1, b=-1:</DIV>
<DIV>exp(-x) + 2*exp(-2x) + 2^4*exp(-4x)*x^2/2! + 2^9*exp(-8x)*x^3/3!
+...+</DIV>
<DIV> 2^(n^2)*exp(-2^n*x)/n! +...</DIV>
<DIV>
<DIV>= 1 + x + 3^2*x^2/2! + 7^3*x^3/3! + 15^4*x^4/4! +...+ </DIV>
<DIV> (2^n-1)^n*x^n/n! +...</DIV></DIV>
<DIV> </DIV></DIV>
<DIV>Example of (5): F(x) = 1+x+x^2, q=2, m=1, b=0: </DIV>
<DIV>Sum_{n>=0} log( 1 + 2^n*x + 2^(2n)*x^2 )^n / n! =
<DIV> Sum_{n>=0} T(2^n, n) * x^n
</DIV></DIV></DIV>
<DIV>where T(2^n,n) = trinomial coefficient of x^n in (1+x+x^2)^(2^n).
</DIV>
<DIV> </DIV>
<DIV>And many other examples can be given. </DIV>
<DIV> </DIV>
<DIV>So the question arises, does (5) have useful
applications? </DIV>
<DIV>Does it offer a g.f. for some significant sequences in OEIS?
</DIV>
<DIV> Paul </DIV>
<DIV> </DIV>
<DIV> </DIV>
<DIV>On Sun, 30 Dec 2007 21:03:39 -0500 <A
href="mailto:pauldhanna@juno.com">pauldhanna@juno.com</A> writes:</DIV>
<BLOCKQUOTE dir=ltr
style="PADDING-LEFT: 10px; MARGIN-LEFT: 10px; BORDER-LEFT: #000000 2px solid">
<DIV>Seqfans, </DIV>
<DIV> Recall the identity: </DIV>
<DIV>(2) Sum_{n>=0} log(1 + q^n*x)^n/n! = Sum_{n>=0}
C(q^n,n)*x^n. </DIV>
<DIV> </DIV>
<DIV>From this, I found the more general statements: </DIV>
<DIV> </DIV>
<DIV>(3) Sum_{n>=0} m^n*log(1 + q^n*x)^n/n! = Sum_{n>=0}
C(m*q^n,n)*x^n.</DIV>
<DIV> </DIV>
<DIV>
<DIV>(4) Sum_{n>=0} m^n * (1 + q^n*x)^b * log(1 + q^n*x)^n/n!
= </DIV>
<DIV> Sum_{n>=0} C(m*q^n + b,
n)*x^n.</DIV>
<DIV> </DIV>
<DIV>Identity (4) is very interesting ... I wonder if it leads to other
results? </DIV>
<DIV>It certainly can lead to many significant sequences! </DIV>
<DIV> </DIV>
<DIV>What I would really like is for formula (4) to allow the
g.f. </DIV>
<DIV> A(x,m,b) = Sum_{n>=0} C(m*q^n + b, n)*x^n
</DIV>
<DIV>to be manipulated to solve some functional equation ... </DIV>
<DIV> </DIV>
<DIV>Any ideas along these lines from anyone?</DIV>
<DIV> Paul</DIV></DIV></BLOCKQUOTE>
<DIV> </DIV></BLOCKQUOTE>
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