%I #5 Jul 13 2018 08:20:54

%S 1,1,6,54,640,9380,163576,3305484,75915708,1952409954,55573310936,

%T 1734182983962,58863621238500,2159006675844616,85088103159523296,

%U 3585740237981536700,160894462797493581048,7658326127259130753070

%N A diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108).

%C Triangle A158835 transforms this sequence into A158832, the next diagonal in A158825.

%e Table of coefficients in the i-th iteration of x*Catalan(x):

%e (1);

%e 1,(1),2,5,14,42,132,429,1430,4862,16796,58786,208012,...;

%e 1,2,(6),21,80,322,1348,5814,25674,115566,528528,2449746,...;

%e 1,3,12,(54),260,1310,6824,36478,199094,1105478,6227712,...;

%e 1,4,20,110,(640),3870,24084,153306,993978,6544242,43652340,...;

%e 1,5,30,195,1330,(9380),67844,500619,3755156,28558484,...;

%e 1,6,42,315,2464,19852,(163576),1372196,11682348,100707972,...;

%e 1,7,56,476,4200,38052,351792,(3305484),31478628,303208212,...;

%e 1,8,72,684,6720,67620,693048,7209036,(75915708),807845676,...;

%e 1,9,90,945,10230,113190,1273668,14528217,167607066,(1952409954),...; ...

%e where terms in parenthesis form the initial terms of this sequence.

%t nmax = 18;

%t g[x_] := Module[{y}, Expand[Normal[(1 - Sqrt[1 - 4*y])/2 + O[y]^(nmax+2)] /. y -> x][[1 ;; nmax+1]]];

%t T = Table[Nest[g, x, n] // CoefficientList[#, x]& // Rest, {n, 1, nmax+1}];

%t Prepend[Diagonal[T, 1], 1] (* _Jean-François Alcover_, Jul 13 2018 *)

%o (PARI) {a(n)=local(F=serreverse(x-x^2+O(x^(n+2))),G=x); for(i=1,n-1,G=subst(F,x,G));polcoeff(G,n)}

%Y Cf. A158825, A158832, A158833, A158834.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Mar 28 2009