Paper IX visual guide · Visual 16 of 23

Compute the reciprocal before estimating it

For width \(N\) and address \(j\), the normalized source and its reciprocal are

\[ G_{N,j}(t)=j!\sum_{n\ge0}\frac{t^{Nn}}{(Nn+j)!},\qquad G_{N,j}(t)^{-1}=\sum_{n\ge0}C_{N,j}(n)\frac{t^{Nn}}{(Nn)!}. \]

Multiplying the two series and clearing the factorials gives an exact triangular recurrence:

\[ C_{N,j}(0)=1,\qquad C_{N,j}(n)=-\frac{\displaystyle\sum_{r=0}^{n-1} \binom{Nn+j}{Nr}C_{N,j}(r)}{\binom{Nn+j}{Nn}}. \]

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Exact reciprocal coefficients at rank four — Original figure from the retained vector PDF.

Reading the figure

The four rows correspond to addresses zero through three. Columns give the normalized reciprocal at rows zero, one and two as exact rational values. They are retained from the worked-example result record.

Exact reciprocal rows
Address	C(0)	C(1)	C(2)
0	1	-1	69
1	1	-1/5	121/45
2	1	-1/15	13/45
3	1	-1/35	59/1155

Selectable labels from the original PDF

Native PDF text retains the figure’s extraction order. Equations and data tables above provide the reading order.

address j C(0) C(1) C(2) 0 1 -1 69 1 1 -1/5 121/45 2 1 -1/15 13/45 3 1 -1/35 59/1155

Original vector figure PDF · Full-size image

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The table is for \(N=4\). The first two address-zero rows illustrate the computation without any valuation estimate:

\[ C_{4,0}(1)=-1,\qquad C_{4,0}(2)=-\bigl(1+\binom84(-1)\bigr)=69. \]

At address one, the same recurrence yields \(C_{4,1}(1)=-1/5\) and \(C_{4,1}(2)=121/45\). These are exact rational coefficients, not rounded approximations. The second implementation in the supplied script inverts the ordinary power series first, then multiplies by \((Nn)!\). Agreement between the two constructions checks the factorial placement.

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Source and conversion record

The full edition has 23 sections and 20 figures, including eight worked examples. The older visual README describes 15 sections and 12 figures. Original wording, equations and finite-check scope are retained. Descriptions and data tables are reading aids. This conversion renders existing figure PDFs without rebuilding TeX, running the mathematical generators, auditing proofs or compiling Lean.

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