[seqfan] Re: A281181 - Need Formula for Terms

[Paul Hanna]

Seqfans,
   The e.g.f. of A281181 is one of a family of nice functions that I wish
to describe here.
A281181 begins (zero-valued coefficients omitted):
[1, 1, 13, 493, 37369, 4732249, 901188997, ...]

Notice that the e.g.f. of A281181 satisfies:
  C(x) = cosh( Integral C(x)^3 dx )
as well as
  C(x)^5 = d/dx Series_Reversion( Integral C(i*x)^5 dx ).

This generalizes to a nice family of interrelated functions,
characterized by the following observation.

OBSERVATION.
If
  C(x) = cosh( Integral C(x)^m dx ),
then
  C(x)^(2*m-1) = d/dx Series_Reversion( Integral C(i*x)^(2*m-1) dx ).

I give some examples below (given in quick PARI code, which I hope is
intelligible).
I hope to submit these sequences as a table sometime.

In case you are interested ...
   Paul

EXAMPLES.
----------------------------------------------------------------------

\\ CASE m=4:  C(x) = cosh( Integral C(x)^4 dx )  begins:

C=1; for(i=1,31, C = cosh( intformal( C^4 +O(x^31)))); Vec(serlaplace(C))
\\ [1, 0, 1, 0, 17, 0, 865, 0, 88865, 0, 15335425, 0, 3993275825, ...]


\\ ( d/dx Series_Reversion( Integral C(i*x)^k dx ) )^(1/k) for k=3..7

Vec(serlaplace( deriv( serreverse( intformal( subst(C,x,I*x)^3 )))^(1/3)))
\\ [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...]

Vec(serlaplace( deriv( serreverse( intformal( subst(C,x,I*x)^4 )))^(1/4)))
\\ [1, 0, 1, 0, 5, 0, 61, 0, 1385, 0, 50521, 0, 2702765, 0, 199360981, ...]

Vec(serlaplace( deriv( serreverse( intformal( subst(C,x,I*x)^5 )))^(1/5)))
\\ [1, 0, 1, 0, 9, 0, 225, 0, 11025, 0, 893025, 0, 108056025, ...]

Vec(serlaplace( deriv( serreverse( intformal( subst(C,x,I*x)^6 )))^(1/6)))
\\ [1, 0, 1, 0, 13, 0, 493, 0, 37369, 0, 4732249, 0, 901188997, ...]

Vec(serlaplace( deriv( serreverse( intformal( subst(C,x,I*x)^7 )))^(1/7)))
\\ [1, 0, 1, 0, 17, 0, 865, 0, 88865, 0, 15335425, 0, 3993275825, ...]


----------------------------------------------------------------------

\\ CASE m=5:  C(x) = cosh( Integral C(x)^5 dx )  begins:

C=1; for(i=1,31, C = cosh( intformal( C^5 +O(x^31)))); Vec(serlaplace(C))
[1, 0, 1, 0, 21, 0, 1341, 0, 173961, 0, 38032281, 0, 12572222301, ...]


\\ ( d/dx Series_Reversion( Integral C(i*x)^k dx ) )^(1/k) for k=4..10

Vec(serlaplace( deriv( serreverse( intformal( subst(C,x,I*x)^4 )))^(1/4)))
\\ [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...]

Vec(serlaplace( deriv( serreverse( intformal( subst(C,x,I*x)^5 )))^(1/5)))
\\ [1, 0, 1, 0, 5, 0, 61, 0, 1385, 0, 50521, 0, 2702765, 0, 199360981, ...]

Vec(serlaplace( deriv( serreverse( intformal( subst(C,x,I*x)^6 )))^(1/6)))
\\ [1, 0, 1, 0, 9, 0, 225, 0, 11025, 0, 893025, 0, 108056025, ...]

Vec(serlaplace( deriv( serreverse( intformal( subst(C,x,I*x)^7 )))^(1/7)))
\\ [1, 0, 1, 0, 13, 0, 493, 0, 37369, 0, 4732249, 0, 901188997, ...]

Vec(serlaplace( deriv( serreverse( intformal( subst(C,x,I*x)^8 )))^(1/8)))
\\ [1, 0, 1, 0, 17, 0, 865, 0, 88865, 0, 15335425, 0, 3993275825, ...]

Vec(serlaplace( deriv( serreverse( intformal( subst(C,x,I*x)^9 )))^(1/9)))
\\ [1, 0, 1, 0, 21, 0, 1341, 0, 173961, 0, 38032281, 0, 12572222301, ...]

Vec(serlaplace( deriv( serreverse( intformal( subst(C,x,I*x)^10 )))^(1/10)))
\\ [1, 0, 1, 0, 25, 0, 1921, 0, 301105, 0, 79715041, 0, 31953352585, ...]


----------------------------------------------------------------------

\\ CASE m=6:  C(x) = cosh( Integral C(x)^6 dx )  begins:

C=1; for(i=1,31, C = cosh( intformal( C^6 +O(x^31)))); Vec(serlaplace(C))
\\ [1, 0, 1, 0, 25, 0, 1921, 0, 301105, 0, 79715041, 0, 31953352585, ...]

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[END]