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On pg. 85 of The Rise and Development of Theory of Series up to the Early 1820s by Ferraro is a series soln. of

$$ d^2z/z = -x^2dx^2 $$

related to the reputed first appearance of a Riccati-type eqn., addressed by the Bernoulli brothers,

$$dy = y^2dx+x^2dx $$

where $y=-\frac{dz}{zdx}$, an early appearance of the so-called Cole-Hopf transformation.

The series is

$$z= 1-x^4/(3 \cdot 4)+x^8/(3 \cdot 4 \cdot 7 \cdot 8) - x^{12}/(3 \cdot 4 \cdot 7 \cdot 8 \cdot 11 \cdot 12) + .... $$

and the denominators are $d_n \rightarrow (1,12,672,88704...)$.

Looking for connections to enumerative combinatorics, I found the first three terms of the series in the 5th row of the square table rep of OEIS A060638, related to the combinatorics of zonotopes.

Does this relationship to the OEIS entry persist, at least empirically, for higher order terms?

(Related: A214916 and a cross-posting as an MSE-Q with a Bessel fct. rep for z.)

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  • $\begingroup$ So the first check is whether $T(8,4)=88704$? $\endgroup$
    – Pietro Majer
    Commented Jan 24, 2017 at 23:19
  • $\begingroup$ @PietroMajer, yep. $\endgroup$
    – Tom Copeland
    Commented Jan 24, 2017 at 23:35
  • $\begingroup$ @PietroMajer, clearly, the series satisfies the recursion relation in A214916 since $-z^{''}=x^2z $, for a periodic subset of that entry (express $z$ as an e.g.f. as in the draft for the entry), so we need a continuation of T only for comparison. $\endgroup$
    – Tom Copeland
    Commented Jan 25, 2017 at 1:16
  • $\begingroup$ The coefficients of $y$ might also be interesting. $\endgroup$
    – Tom Copeland
    Commented Jan 25, 2017 at 1:53
  • $\begingroup$ For more on early Riccati equations, see "The origins of Euler’s early work on continued fractions" by Rosanna Cretney $\endgroup$
    – Tom Copeland
    Commented Oct 2, 2020 at 3:56

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