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About Some Integer Series
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Pascal's Triangle: figurate
numbers and more
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The famous binomial
theorem/combinatorial/probabilistic numbers.
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Euler's Triangle
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Definitions
include: the formula (based on Pascal's Triangle), an internal formula (based
on previous values), and how the triangle represents the count of the possible
number of sets that have a certain amount of ascents within them.
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Euler's Triangle: a doubly enlightening grasp of the accumulation to
values of powers--as Euler/Pascal Weightings and as Seed Numbers accumulating
to Power Series
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The triangle
relates to the figurate numbers of Pascal's Triangle. This may be what Fermat
sensed: the cumulative nature of figurate numbers is interior to and
describes accumulation within power series.
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Also, working backward from the
power/shell series, a differences of series realize each row of Euler's Triangle as an end result for each power series. This means that each row of Euler's numbers are
'seed
numbers' for direct, continuous accumulation to shells, powers and sums of
powers.
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About Sieved Integer Series
A chart of accumulations that are
the powers
A
comprehensive pattern of series with deletions creates any x^n series
by accumulation of accumulations--may be what Fermat saw. It is a source of
binomial theory.
A power value accumulates by
addition
Formulas defining the accumulations
of powers
These formulas
are drawn from the chart of accumulations that has deletions in the series.
How to Mathematically Not Count Non-Stuff
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