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1: 28.2 Definitions and Basic Properties
The general solution of (28.2.16) is ν = ± ν ^ + 2 n , where n . …
28.2.19 q c 2 n + 2 - ( a - ( ν + 2 n ) 2 ) c 2 n + q c 2 n - 2 = 0 , n .
28.2.23 a n ( 0 ) = n 2 , n = 0 , 1 , 2 , ,
28.2.24 b n ( 0 ) = n 2 , n = 1 , 2 , 3 , .
§28.2(vi) Eigenfunctions
2: 1.1 Special Notation
x , y real variables.
j , k , integers.
m , n nonnegative integers, unless specified otherwise.
3: 32.1 Special Notation
m , n integers.
4: 24.6 Explicit Formulas
24.6.1 B 2 n = k = 2 2 n + 1 ( - 1 ) k - 1 k ( 2 n + 1 k ) j = 1 k - 1 j 2 n ,
24.6.2 B n = 1 n + 1 k = 1 n j = 1 k ( - 1 ) j j n ( n + 1 k - j ) / ( n k ) ,
24.6.3 B 2 n = k = 1 n ( k - 1 ) ! k ! ( 2 k + 1 ) ! j = 1 k ( - 1 ) j - 1 ( 2 k k + j ) j 2 n .
24.6.4 E 2 n = k = 1 n 1 2 k - 1 j = 1 k ( - 1 ) j ( 2 k k - j ) j 2 n ,
24.6.9 B n = k = 0 n 1 k + 1 j = 0 k ( - 1 ) j ( k j ) j n ,
5: 27.19 Methods of Computation: Factorization
§27.19 Methods of Computation: Factorization
Techniques for factorization of integers fall into three general classes: Deterministic algorithms, Type I probabilistic algorithms whose expected running time depends on the size of the smallest prime factor, and Type II probabilistic algorithms whose expected running time depends on the size of the number to be factored. … Type II probabilistic algorithms for factoring n rely on finding a pseudo-random pair of integers ( x , y ) that satisfy x 2 y 2 ( mod n ) . …As of January 2009 the snfs holds the record for the largest integer that has been factored by a Type II probabilistic algorithm, a 307-digit composite integer. …The largest composite numbers that have been factored by other Type II probabilistic algorithms are a 63-digit integer by cfrac, a 135-digit integer by mpqs, and a 182-digit integer by gnfs. …
6: 27.6 Divisor Sums
27.6.1 d | n λ ( d ) = { 1 , n  is a square , 0 , otherwise .
27.6.2 d | n μ ( d ) f ( d ) = p | n ( 1 - f ( p ) ) , n > 1 .
27.6.6 d | n ϕ k ( d ) ( n d ) k = 1 k + 2 k + + n k ,
27.6.7 d | n μ ( d ) ( n d ) k = J k ( n ) ,
27.6.8 d | n J k ( d ) = n k .
7: 26.9 Integer Partitions: Restricted Number and Part Size
§26.9 Integer Partitions: Restricted Number and Part Size
§26.9(i) Definitions
Table 26.9.1: Partitions p k ( n ) .
n k
§26.9(ii) Generating Functions
§26.9(iii) Recurrence Relations
8: 14.27 Zeros
  • (a)

    μ < 0 , μ , ν , and sin ( ( μ - ν ) π ) and sin ( μ π ) have opposite signs.

  • (b)

    μ , ν , μ + ν < 0 , and ν is odd.

  • 9: 26.12 Plane Partitions
    A plane partition, π , of a positive integer n , is a partition of n in which the parts have been arranged in a 2-dimensional array that is weakly decreasing (nonincreasing) across rows and down columns. …
    26.12.3 B ( r , s , t ) = { ( h , j , k ) |  1 h r , 1 j s , 1 k t } .
    26.12.9 ( h = 1 r j = 1 s h + j + t - 1 h + j - 1 ) 2 ;
    26.12.13 h = 1 r j = 1 r h + j + t - 1 h + j - 1 ;
    26.12.14 h = 1 r j = 1 r + 1 h + j + t - 1 h + j - 1 .
    10: 24.1 Special Notation
    j , k , , m , n integers, nonnegative unless stated otherwise.
    Unless otherwise noted, the formulas in this chapter hold for all values of the variables x and t , and for all nonnegative integers n . …