About the Project

cash app phone number +1(888‒481‒4477)

AdvancedHelp

(0.014 seconds)

11—20 of 61 matching pages

11: 24.11 Asymptotic Approximations
24.11.1 ( 1 ) n + 1 B 2 n 2 ( 2 n ) ! ( 2 π ) 2 n ,
24.11.2 ( 1 ) n + 1 B 2 n 4 π n ( n π e ) 2 n ,
12: 3.11 Approximation Techniques
3.11.28 S = j = 1 J ( f ( x j ) p n ( x j ) ) 2 .
Here x j , j = 1 , 2 , , J , is a given set of distinct real points and J n + 1 . …
3.11.32 j = 1 J w ( x j ) ( f ( x j ) Φ n ( x j ) ) 2 ,
3.11.34 X k = j = 1 J w ( x j ) ϕ k ( x j ) ϕ ( x j ) ,
3.11.35 F k = j = 1 J w ( x j ) f ( x j ) ϕ k ( x j ) .
13: 24.9 Inequalities
24.9.6 5 π n ( n π e ) 2 n > ( 1 ) n + 1 B 2 n > 4 π n ( n π e ) 2 n ,
24.9.8 2 ( 2 n ) ! ( 2 π ) 2 n 1 1 2 β 2 n ( 1 ) n + 1 B 2 n 2 ( 2 n ) ! ( 2 π ) 2 n 1 1 2 2 n
24.9.10 4 n + 1 ( 2 n ) ! π 2 n + 1 > ( 1 ) n E 2 n > 4 n + 1 ( 2 n ) ! π 2 n + 1 1 1 + 3 1 2 n .
14: 26.7 Set Partitions: Bell Numbers
26.7.6 B ( n + 1 ) = k = 0 n ( n k ) B ( k ) .
15: 26.8 Set Partitions: Stirling Numbers
26.8.20 s ( n + 1 , k + 1 ) = n ! j = k n ( 1 ) n j j ! s ( j , k ) ,
26.8.22 S ( n , k ) = k S ( n 1 , k ) + S ( n 1 , k 1 ) ,
26.8.25 S ( n + 1 , k + 1 ) = j = k n ( n j ) S ( j , k ) ,
26.8.30 j = k n s ( n + 1 , j + 1 ) n j k = s ( n , k ) .
For asymptotic approximations for s ( n + 1 , k + 1 ) and S ( n , k ) that apply uniformly for 1 k n as n see Temme (1993) and Temme (2015, Chapter 34). …
16: DLMF Project News
error generating summary
17: 24.4 Basic Properties
24.4.7 k = 1 m k n = B n + 1 ( m + 1 ) B n + 1 n + 1 ,
24.4.11 k = 1 ( k , m ) = 1 m k n = 1 n + 1 j = 1 n + 1 ( n + 1 j ) ( p | m ( 1 p n j ) B n + 1 j ) m j .
24.4.26 E n ( 0 ) = E n ( 1 ) = 2 n + 1 ( 2 n + 1 1 ) B n + 1 , n > 0 .
24.4.33 E 2 n ( 1 6 ) = E 2 n ( 5 6 ) = 1 + 3 2 n 2 2 n + 1 E 2 n .
18: 26.14 Permutations: Order Notation
It is also equal to the number of permutations in 𝔖 n with exactly k + 1 weak excedances. …
26.14.8 n k = ( k + 1 ) n 1 k + ( n k ) n 1 k 1 , n 2 ,
26.14.16 n 2 = 3 n ( n + 1 ) 2 n + ( n + 1 2 ) , n 1 .
19: 24.5 Recurrence Relations
24.5.6 k = 2 n ( n k 2 ) B k k = 1 ( n + 1 ) ( n + 2 ) B n + 1 , n = 2 , 3 , ,
24.5.7 k = 0 n ( n k ) B k n + 2 k = B n + 1 n + 1 , n = 1 , 2 , ,
20: 24.13 Integrals
24.13.4 0 1 / 2 B n ( t ) d t = 1 2 n + 1 2 n B n + 1 n + 1 ,
24.13.5 1 / 4 3 / 4 B n ( t ) d t = E n 2 2 n + 1 .
24.13.8 0 1 E n ( t ) d t = 2 E n + 1 ( 0 ) n + 1 = 4 ( 2 n + 2 1 ) ( n + 1 ) ( n + 2 ) B n + 2 ,
24.13.9 0 1 / 2 E 2 n ( t ) d t = E 2 n + 1 ( 0 ) 2 n + 1 = 2 ( 2 2 n + 2 1 ) B 2 n + 2 ( 2 n + 1 ) ( 2 n + 2 ) ,