0,3

Row 0 has 1 term; row n (n>=1) has ceil(n/2) terms. Row sums yield the Catalan numbers (A000108). Column 0 yields A135307. T(2n+1,n)=binom(2n,n)/(n+1) (the Catalan numbers, A000108). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 15 2007

A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.

T(n,k)=(1/n)binom(n,k)Sum[(-1)^(j-k)*binom(n-k,j-k)binom(2n-3j,n-j+1),j=k..floor((n-1)/2)]. G.f.=G=G(t,z) satisfies zG^3 - [(1-t)z+1]G^2 + [1+2(1-t)z]G - (1-t)z=0. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 15 2007

Triangle begins:

1

1

2

4 1

9 5

23 17 2

63 54 15

178 177 69 5

514 594 273 49

...

T(4,1)=5 because we have U(UDDU)DUD, U(UDDU)UDD, UU(UDDU)DD, UDU(UDDU)D and UUD(UDDU)D (the UDDU's are shown between parentheses).

A135306 := proc(n, k) if n =0 then 1 ; else add((-1)^(j-k)*binomial(n-k, j-k)*binomial(2*n-3*j, n-j+1), j=k..floor((n-1)/2)) ; %*binomial(n, k)/n ; fi ; end: for n from 0 to 20 do for k from 0 to max(0, (n-1)/2) do printf("%a, ", A135306(n, k)) ; od: od: - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Dec 08 2007

T:=proc(n, k) options operator, arrow: binomial(n, k)*(sum((-1)^(j-k)*binomial(n-k, j-k)*binomial(2*n-3*j, n-j+1), j=k..floor((1/2)*n-1/2)))/n end proc: 1; for n to 13 do seq(T(n, k), k=0..ceil((n-2)*1/2)) end do; # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 15 2007

Cf. A000108, A135307.

Sequence in context: A163240 A091958 A116424 this_sequence A102405 A114506 A114848

Adjacent sequences: A135303 A135304 A135305 this_sequence A135307 A135308 A135309

nonn,tabf

N. J. A. Sloane (njas(AT)research.att.com), Dec 07 2007

More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl) and Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 08 2007