2,1

Sum of entries in row n is n! (A000142(n)).

T(n,n-1)=A092186(n).

T(n,1)=A152875(n).

Sum(k*T(n,k),k=1..n-1)=2*A077613(n).

T(2n,k)=(n!)^2*a(n,k), where a(n,k) is the number of lattice paths from (0,0) to (n,n) with steps N=(0,1) and E=(1,0) and having k turns;

a(n,k)=2*binom(n-1,k/2-1)*binom(n-1,k/2) if k even;

a(n,k)=2*[binom(n-1,(k-1)/2)]^2 if k odd.

T(2n+1,k)=n!*(n+1)!*b(n,k), where b(n,k) is the number of lattice paths from (0,0) to (n,n+1) with steps N=(0,1) and E=(1,0) and having k turns;

b(n,k)=binom(n,k/2)*binom(n-1,k/2-1)+binom(n,k/2-1)*binom(n-1,k/2)=[binom(n,k/2)]^2*k(2n-k+1)/[n(2n-k+2)] if k even;

b(n,k)=2*binom(n,(k-1)/2)*binom(n-1,(k-1)/2) if k odd.

T(4,3)=8 because we have 1243, 1423, 4132, 4312, 2134, 2314, 3241 and 3421.

Triangle starts:

2;

4,2;

8,8,8;

24,36,48,12;

72,144,288,144,72

ae := proc (n, k) if `mod`(k, 2) = 0 then 2*factorial(n)^2*binomial(n-1, (1/2)*k-1)*binomial(n-1, (1/2)*k) else 2*factorial(n)^2*binomial(n-1, (1/2)*k-1/2)^2 end if end proc: ao := proc (n, k) if `mod`(k, 2) = 0 then factorial(n)*factorial(n+1)*(binomial(n, (1/2)*k)*binomial(n-1, (1/2)*k-1)+binomial(n, (1/2)*k-1)*binomial(n-1, (1/2)*k)) else 2*factorial(n)*factorial(n+1)*binomial(n, (1/2)*k-1/2)*binomial(n-1, (1/2)*k-1/2) end if end proc: T := proc (n, k) if `mod`(n, 2) = 0 then ae((1/2)*n, k) else ao((1/2)*n-1/2, k) end if end proc: for n from 2 to 10 do seq(T(n, k), k = 1 .. n-1) end do; # yields sequence in triangular form

A000142, A092186, A077613

Sequence in context: A163897 A113477 A129178 this_sequence A065286 A068217 A114593

Adjacent sequences: A152871 A152872 A152873 this_sequence A152875 A152876 A152877

nonn,tabl

Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 15 2008