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1: 26.3 Lattice Paths: Binomial Coefficients
§26.3 Lattice Paths: Binomial Coefficients
§26.3(i) Definitions
For numerical values of ( m n ) and ( m + n n ) see Tables 26.3.1 and 26.3.2. …
§26.3(ii) Generating Functions
§26.3(iv) Identities
2: 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 ,
3: 24.5 Recurrence Relations
24.5.1 k = 0 n 1 ( n k ) B k ( x ) = n x n 1 , n = 2 , 3 , ,
24.5.3 k = 0 n 1 ( n k ) B k = 0 , n = 2 , 3 , ,
24.5.4 k = 0 n ( 2 n 2 k ) E 2 k = 0 , n = 1 , 2 , ,
24.5.5 k = 0 n ( n k ) 2 k E n k + E n = 2 .
a n = k = 0 n ( n k ) b n k k + 1 ,
4: 26.21 Tables
§26.21 Tables
Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients ( m n ) for m up to 50 and n up to 25; extends Table 26.4.1 to n = 10 ; tabulates Stirling numbers of the first and second kinds, s ( n , k ) and S ( n , k ) , for n up to 25 and k up to n ; tabulates partitions p ( n ) and partitions into distinct parts p ( 𝒟 , n ) for n up to 500. … It also contains a table of Gaussian polynomials up to [ 12 6 ] q . Goldberg et al. (1976) contains tables of binomial coefficients to n = 100 and Stirling numbers to n = 40 .
5: 24.14 Sums
24.14.1 k = 0 n ( n k ) B k ( x ) B n k ( y ) = n ( x + y 1 ) B n 1 ( x + y ) ( n 1 ) B n ( x + y ) ,
24.14.2 k = 0 n ( n k ) B k B n k = ( 1 n ) B n n B n 1 .
24.14.3 k = 0 n ( n k ) E k ( h ) E n k ( x ) = 2 ( E n + 1 ( x + h ) ( x + h 1 ) E n ( x + h ) ) ,
24.14.5 k = 0 n ( n k ) E k ( h ) B n k ( x ) = 2 n B n ( 1 2 ( x + h ) ) ,
24.14.6 k = 0 n ( n k ) 2 k B k E n k = 2 ( 1 2 n 1 ) B n n E n 1 .
6: 17.2 Calculus
§17.2(ii) Binomial Coefficients
17.2.30 [ n m ] q = [ m + n 1 m ] q ( 1 ) m q m n ( m 2 ) ,
§17.2(iii) Binomial Theorem
In the limit as q 1 , (17.2.35) reduces to the standard binomial theorem … When a = q m + 1 , where m is a nonnegative integer, (17.2.37) reduces to the q -binomial series …
7: 26.5 Lattice Paths: Catalan Numbers
26.5.1 C ( n ) = 1 n + 1 ( 2 n n ) = 1 2 n + 1 ( 2 n + 1 n ) = ( 2 n n ) ( 2 n n 1 ) = ( 2 n 1 n ) ( 2 n 1 n + 1 ) .
26.5.5 C ( n + 1 ) = k = 0 n / 2 ( n 2 k ) 2 n 2 k C ( k ) .
8: 17.3 q -Elementary and q -Special Functions
17.3.2 E q ( x ) = n = 0 ( 1 q ) n q ( n 2 ) x n ( q ; q ) n = ( ( 1 q ) x ; q ) .
17.3.7 β n ( x , q ) = ( 1 q ) 1 n r = 0 n ( 1 ) r ( n r ) r + 1 ( 1 q r + 1 ) q r x .
17.3.8 A m , s ( q ) = q ( s m 2 ) + ( s 2 ) j = 0 s ( 1 ) j q ( j 2 ) [ m + 1 j ] q ( 1 q s j ) m ( 1 q ) m .
17.3.9 a m , s ( q ) = q ( s 2 ) ( 1 q ) s ( q ; q ) s j = 0 s ( 1 ) j q ( j 2 ) [ s j ] q ( 1 q s j ) m ( 1 q ) m .
9: 26.17 The Twelvefold Way
Table 26.17.1: The twelvefold way.
elements of N elements of K f unrestricted f one-to-one f onto
unlabeled labeled ( k + n 1 n ) ( k n ) ( n 1 n k )
10: 12.13 Sums
12.13.2 U ( a , x + y ) = e 1 2 x y 1 4 y 2 m = 0 ( a 1 2 m ) y m U ( a + m , x ) ,
12.13.3 V ( a , x + y ) = e 1 2 x y + 1 4 y 2 m = 0 ( a 1 2 m ) y m V ( a m , x ) ,
12.13.5 U ( a , x cos t + y sin t ) = e 1 4 ( x sin t y cos t ) 2 m = 0 ( a 1 2 m ) ( tan t ) m U ( m + a , x ) U ( m 1 2 , y ) , a 1 2 , 0 t 1 4 π .