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11: 26.5 Lattice Paths: Catalan Numbers
§26.5 Lattice Paths: Catalan Numbers
►§26.5(i) Definitions
► is the Catalan number. … ►§26.5(ii) Generating Function
… ►§26.5(iii) Recurrence Relations
…12: Nico M. Temme
…
►On the occasion of his retirement in 2005 he was awarded the decoration Knight in the Order of the Dutch Lion, issued by the King of the Netherlands.
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13: 4.19 Maclaurin Series and Laurent Series
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►In (4.19.3)–(4.19.9), are the Bernoulli numbers and are the Euler numbers (§§24.2(i)–24.2(ii)).
►
4.19.3
,
►
4.19.4
,
►
4.19.5
,
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4.19.6
,
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14: 26.14 Permutations: Order Notation
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►As an example, is an element of The inversion number is the number of pairs of elements for which the larger element precedes the smaller:
…
►
►The Eulerian number, denoted , is the number of permutations in with exactly descents.
…The Eulerian number
is equal to the number of permutations in with exactly excedances.
…
►
§26.14(iii) Identities
…15: 26.7 Set Partitions: Bell Numbers
§26.7 Set Partitions: Bell Numbers
►§26.7(i) Definitions
… ►§26.7(ii) Generating Function
… ►§26.7(iii) Recurrence Relation
… ►§26.7(iv) Asymptotic Approximation
…16: Publications
…
►
D. W. Lozier (2003)
The NIST Digital Library of Mathematical Functions Project,
Annals of Mathematics and Artificial Intelligence—Special Issue on Mathematical Knowledge Management,
Vol. 38, Nos. 1–3, pp. 105–119.
►
B. R. Miller and A. Youssef (2003)
Technical Aspects of the Digital Library of Mathematical Functions,
Annals of Mathematics and Artificial Intelligence—Special Issue on Mathematical Knowledge Management,
Vol. 38, Nos. 1–3, pp. 121–136.
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17: 26.8 Set Partitions: Stirling Numbers
§26.8 Set Partitions: Stirling Numbers
►§26.8(i) Definitions
… ► … ►§26.8(v) Identities
… ►§26.8(vi) Relations to Bernoulli Numbers
…18: 1.2 Elementary Algebra
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►The arithmetic mean of
numbers
is
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►If is a nonzero real number, then the weighted mean
of nonnegative numbers
, and positive numbers
with
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►Numerical methods and issues for solution of (1.2.61) appear in §§3.2(i) to 3.2(iii).
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►Numerical methods and issues for solution of (1.2.72) appear in §§3.2(iv) to 3.2(vii).
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►The diagonal elements are not necessarily distinct, and the number of identical (degenerate) diagonal elements is the multiplicity of that specific eigenvalue.
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19: 24.19 Methods of Computation
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