%I A102625
%S A102625 1,1,2,3,6,6,15,30,36,24,105,210,270,240,120,945,1890,2520,2520,1800,
%T A102625 720,10395,20790,28350,30240,25200,15120,5040,135135,270270,374220,
%U A102625 415800,378000,272160,141120,40320,2027025,4054050,5675670,6486480
%N A102625 Triangle read by rows: T(n,k) is the sum of the weights of all vertices 
               labeled k at depth n in the Catalan tree (1<=k<=n+1, n>=0).
%C A102625 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).
%C A102625 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
%D A102625 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.
%F A102625 T(n, k)=k(2n-k+1)!/[2^(n-k+1)*(n-k+1)! ] (1<=k<=n+1).
%F A102625 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
%F A102625 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
%F A102625 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
%e A102625 Triangle starts:
%e A102625 1;
%e A102625 1,2;
%e A102625 3,6,6;
%e A102625 15,30,36,24;
%p A102625 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
%Y A102625 Cf. A001147, A001286.
%Y A102625 Sequence in context: A129649 A129650 A007894 this_sequence A117777 A049297 
               A056391
%Y A102625 Adjacent sequences: A102622 A102623 A102624 this_sequence A102626 A102627 
               A102628
%K A102625 nonn,tabl
%O A102625 0,3
%A A102625 Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 31 2005