|
| |
|
|
A115791
|
|
Number of different ways to select n elements from five sets of n elements under the precondition of choosing at least one element from each set.
|
|
0
| |
|
|
0, 0, 0, 0, 3125, 97200, 1932805, 31539200, 461828790, 6332578125, 83235183075, 1063505908080, 13327125965725, 164758298214965, 2017489363833125, 24538128923443200, 297028957324770140, 3583456866615114630
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,5
|
|
|
COMMENTS
| The number of different ways to select n elements from five sets of n elements under the precondition of choosing at least one element from each set.
|
|
|
FORMULA
| a(n) = binomial(5*n,n)-5*binomial(4*n,n)+10*binomial(3*n,n)-10*binomial(2*n,n)+5; ; also: a(n)=sum{binomial(n,i)*binomial(n,j)*binomial(n,k)*binomial(n,l)*binomial(n,m)||i,j,k,l,m=1...(n-4),i+j+k+l+m=n}. General formula for N sets with m elements each: the number of different ways to select k elements from j different sets: G(N,m,j,k) = binomial(N,j)*sum(binomial(j,i)*binomial(i*m,k)*(-1)^i*(-1)^j|i=1...j); Recursion formula: G(N,m,j,k) = binomial(N,j)*binomial(j*m,k) - sum(binomial(N-i,j-i)*G(N,m,i,k)|i=1...j-1);
|
|
|
EXAMPLE
| a(6)=binomial(30,6)-5*binomial(24,6)+10*binomial(18,6)-10*binomial(12,6)+5=97200;
|
|
|
MATHEMATICA
| Table[Binomial[5n, n]-5Binomial[4n, n]+10Binomial[3n, n]-10Binomial[2n, n]+5, {n, 20}] (* From Harvey P. Dale, Nov 06 2011 *)
|
|
|
CROSSREFS
| Cf. A115111, A115112, A115246.
Sequence in context: A084649 A169754 A016817 * A016853 A016973 A017045
Adjacent sequences: A115788 A115789 A115790 * A115792 A115793 A115794
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jan 31 2006
|
| |
|
|
