0,4

First column is A001405, second column is A100071, third column is A107231. Row sums are A005773(n+1), diagonal sums are A026003. The inverse Chebyshev transform concerned takes a g.f. g(x)->(1/sqrt(1-4x^2))g(xc(x^2)) where c(x) is the g.f. of A000108. It transforms a(n) to b(n)=sum{k=0..floor(n/2),C(n,k)a(n-2k)}. Then a(n)=sum{k=0..floor(n/2),(n/(n-k))(-1)^k*C(n-k,k)b(n-2k)}.

Triangle read by rows: T(n,k) is the number of paths of length n with steps U=(1,1), D=(1,-1), and H=(1,0), starting at (0,0), staying weakly above the x-axis (i.e. left factors of Motzkin paths) and having k H steps. Example: T(3,1)=6 because we have HUD. HUU, UDH, UHD, UHU, and UUH. Sum(k*T(n,k), k=0..n)=A132894(n). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 07 2007

T(n, k)=C(n, k)C(n-k, floor((n-k)/2)

G.f.=G=G(t,z) satisfies z(1-2z-tz)G^2+(1-2z-tz)G-1=0. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 07 2007

Triangle begins

1;

1,1;

2,2,1;

3,6,3,1;

6,12,12,4,1;

10,30,30,20,5,1;

T:=proc(n, k) options operator, arrow: binomial(n, k)*binomial(n-k, floor((1/2)*n-(1/2)*k)) end proc: for n from 0 to 11 do seq(T(n, k), k=0..n) end do; # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 07 2007

Cf. A132894.

Sequence in context: A134399 A094436 A094441 this_sequence A046726 A082137 A091187

Adjacent sequences: A107227 A107228 A107229 this_sequence A107231 A107232 A107233

easy,nonn,tabl

Paul Barry (pbarry(AT)wit.ie), May 13 2005