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A158831 A diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108).

%I A158831 #5 Jul 13 2018 08:20:54
%S A158831 1,1,6,54,640,9380,163576,3305484,75915708,1952409954,55573310936,
%T A158831 1734182983962,58863621238500,2159006675844616,85088103159523296,
%U A158831 3585740237981536700,160894462797493581048,7658326127259130753070
%N A158831 A diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108).
%C A158831 Triangle A158835 transforms this sequence into A158832, the next diagonal in A158825.
%e A158831 Table of coefficients in the i-th iteration of x*Catalan(x):
%e A158831 (1);
%e A158831 1,(1),2,5,14,42,132,429,1430,4862,16796,58786,208012,...;
%e A158831 1,2,(6),21,80,322,1348,5814,25674,115566,528528,2449746,...;
%e A158831 1,3,12,(54),260,1310,6824,36478,199094,1105478,6227712,...;
%e A158831 1,4,20,110,(640),3870,24084,153306,993978,6544242,43652340,...;
%e A158831 1,5,30,195,1330,(9380),67844,500619,3755156,28558484,...;
%e A158831 1,6,42,315,2464,19852,(163576),1372196,11682348,100707972,...;
%e A158831 1,7,56,476,4200,38052,351792,(3305484),31478628,303208212,...;
%e A158831 1,8,72,684,6720,67620,693048,7209036,(75915708),807845676,...;
%e A158831 1,9,90,945,10230,113190,1273668,14528217,167607066,(1952409954),...; ...
%e A158831 where terms in parenthesis form the initial terms of this sequence.
%t A158831 nmax = 18;
%t A158831 g[x_] := Module[{y}, Expand[Normal[(1 - Sqrt[1 - 4*y])/2 + O[y]^(nmax+2)] /. y -> x][[1 ;; nmax+1]]];
%t A158831 T = Table[Nest[g, x, n] // CoefficientList[#, x]& // Rest, {n, 1, nmax+1}];
%t A158831 Prepend[Diagonal[T, 1], 1] (* _Jean-François Alcover_, Jul 13 2018 *)
%o A158831 (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 A158831 Cf. A158825, A158832, A158833, A158834.
%K A158831 nonn
%O A158831 1,3
%A A158831 _Paul D. Hanna_, Mar 28 2009