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1: 26.12 Plane Partitions
Table 26.12.1: Plane partitions.
n pp ( n ) n pp ( n ) n pp ( n )
3 6 20 75278 37 903 79784
§26.12(iv) Limiting Form
2: 26.5 Lattice Paths: Catalan Numbers
Table 26.5.1: Catalan numbers.
n C ( n ) n C ( n ) n C ( n )
6 132 13 7 42900 20 65641 20420
§26.5(iv) Limiting Forms
26.5.7 lim n C ( n + 1 ) C ( n ) = 4 .
3: Bibliography C
  • B. C. Carlson (1972b) Intégrandes à deux formes quadratiques. C. R. Acad. Sci. Paris Sér. A–B 274 (15 May, 1972, Sér. A), pp. 1458–1461 (French).
  • P. L. Chebyshev (1851) Sur la fonction qui détermine la totalité des nombres premiers inférieurs à une limite donnée. Mem. Ac. Sc. St. Pétersbourg 6, pp. 141–157.
  • R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.
  • M. Colman, A. Cuyt, and J. Van Deun (2011) Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
  • M. D. Cooper, R. H. Jeppesen, and M. B. Johnson (1979) Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.
  • 4: 26.3 Lattice Paths: Binomial Coefficients
    Table 26.3.1: Binomial coefficients ( m n ) .
    m n
    6 1 6 15 20 15 6 1
    Table 26.3.2: Binomial coefficients ( m + n m ) for lattice paths.
    m n
    3 1 4 10 20 35 56 84 120 165
    26.3.4 m = 0 ( m + n m ) x m = 1 ( 1 x ) n + 1 , | x | < 1 .
    §26.3(v) Limiting Form
    5: Errata
    We have also incorporated material on continuous q -Jacobi polynomials, and several new limit transitions. … The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions. …
  • Chapters 8, 20, 36

    Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

  • Table 22.5.4

    Originally the limiting form for sc ( z , k ) in the last line of this table was incorrect ( cosh z , instead of sinh z ).

    sn ( z , k ) tanh z cd ( z , k ) 1 dc ( z , k ) 1 ns ( z , k ) coth z
    cn ( z , k ) sech z sd ( z , k ) sinh z nc ( z , k ) cosh z ds ( z , k ) csch z
    dn ( z , k ) sech z nd ( z , k ) cosh z sc ( z , k ) sinh z cs ( z , k ) csch z

    Reported 2010-11-23.

  • References

    Bibliographic citations were added in §§1.13(v), 10.14, 10.21(ii), 18.15(v), 18.32, 30.16(iii), 32.13(ii), and as general references in Chapters 19, 20, 22, and 23.

  • 6: 26.9 Integer Partitions: Restricted Number and Part Size
    Table 26.9.1: Partitions p k ( n ) .
    n k
    8 0 1 5 10 15 18 20 21 22 22 22
    Equations (26.9.2)–(26.9.3) are examples of closed forms that can be computed explicitly for any positive integer k . …
    §26.9(iv) Limiting Form