0,3

The Catalan tree is defined as follows: the root is labeled 1 and each vertex labeled i has i+1 children labeled 1,2,...,i+1. The weight of a vertex v is the product of all labels on the path from the root to v. Row n contains n+1 terms. Row sums and column 1 yield the double factorials (A001147). T(n,n+1)=(n+1)!, T(n,n)=n(n+1)!/2 (A001286; Lah numbers).

This table counts permutations of the multiset {1,1,2,2,...,n,n} satisfying the condition "the first appearance of i + 1 follows the first appearance of i" by the position of the first appearance of n. Specifically, T(n+1,k) is the number of such permutations for which n first occurs in position 2n+1-k. For example, with n=2 and k=1, T(3,1)=6 counts 121323, 121332, 122313, 122331, 112323, 112332. - David Callan (callan(AT)stat.wisc.edu), Nov 29 2007

S. Lehr, J. Shallit and J. Tromp, On the vector space of the automatic reals, Theoret. Comput. Sci. 163 (1996), no. 1-2, 193-210.

T(n, k)=k(2n-k+1)!/[2^(n-k+1)*(n-k+1)! ] (1<=k<=n+1).

Bivariate G.F. = exp[P(.,t)*x] = D_x {1 - [g(x)/(1+t*g(x)]} = 1 / {(1+g(x))*[1+t*g(x)]^2} , where g(x) = sqrt(1-2*x) - 1 and P(n,t) = sum(k=0,..,n) T(n,k)* t^k . - Tom Copeland (tcjpn(AT)msn.com), Nov 11 2007

Also D_x g(x) = -(1-2*x)^(-1/2) = -exp[x*A001147(.)] = -exp[x *(2*(.)-1)!! ] , so the coefficients of x^n/n! in the expansion of g(x) are -(2*(n-1)-1)!! = -A001147(n-1) for n > 0 . - Tom Copeland (tcjpn(AT)msn.com), Nov 11 2007

See A132382 for an array which is essentially the revert from which this G.F. may be derived and for connections to other arrays. - Tom Copeland (tcjpn(AT)msn.com), Nov 11 2007

Triangle starts:

1;

1,2;

3,6,6;

15,30,36,24;

T:=proc(n, k) if k<=n+1 then k*(2*n-k+1)!/2^(n-k+1)/(n-k+1)! else 0 fi end: for n from 0 to 8 do seq(T(n, k), k=1..n+1) od; # yields sequence in triangular form

Cf. A001147, A001286.

Adjacent sequences: A102622 A102623 A102624 this_sequence A102626 A102627 A102628

Sequence in context: A129649 A129650 A007894 this_sequence A117777 A049297 A056391

nonn,tabl

Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 31 2005