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A158832 Main diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108).

%I A158832 #7 Jul 13 2018 08:19:45
%S A158832 1,2,12,110,1330,19852,351792,7209036,167607066,4357308098,
%T A158832 125219900520,3941126688798,134808743674176,4979127855477336,
%U A158832 197480359402576304,8370550907396970684,377599345119560766534,18061714498169627460982
%N A158832 Main diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108).
%C A158832 Triangle A158835 transforms A158831 into this sequence, where A158831 is the previous diagonal in A158825.
%C A158832 Triangle A158835 transforms this sequence into A158833, the next diagonal in A158825.
%H A158832 Paul D. Hanna, <a href="/A158832/b158832.txt">Table of n, a(n), n = 1..50.</a>
%e A158832 Array of coefficients in the i-th iteration of x*Catalan(x):
%e A158832 (1),1,2,5,14,42,132,429,1430,4862,16796,58786,208012,...;
%e A158832 1,(2),6,21,80,322,1348,5814,25674,115566,528528,2449746,...;
%e A158832 1,3,(12),54,260,1310,6824,36478,199094,1105478,6227712,...;
%e A158832 1,4,20,(110),640,3870,24084,153306,993978,6544242,43652340,...;
%e A158832 1,5,30,195,(1330),9380,67844,500619,3755156,28558484,...;
%e A158832 1,6,42,315,2464,(19852),163576,1372196,11682348,100707972,...;
%e A158832 1,7,56,476,4200,38052,(351792),3305484,31478628,303208212,...;
%e A158832 1,8,72,684,6720,67620,693048,(7209036),75915708,807845676,...;
%e A158832 1,9,90,945,10230,113190,1273668,14528217,(167607066),...;
%e A158832 1,10,110,1265,14960,180510,2212188,27454218,344320262,(4357308098),...; ...
%e A158832 where terms in parenthesis form the initial terms of this sequence.
%t A158832 a[n_] := Module[{x, F, G}, F = InverseSeries[x - x^2 + O[x]^(n+2)]; G = x; For[i = 1, i <= n, i++, G = (F /. x -> G)]; Coefficient[G, x, n]];
%t A158832 Array[a, 18] (* _Jean-François Alcover_, Jul 13 2018, from PARI *)
%o A158832 (PARI) {a(n)=local(F=serreverse(x-x^2+O(x^(n+2))),G=x); for(i=1,n,G=subst(F,x,G));polcoeff(G,n)}
%Y A158832 Cf. A158825, A158831, A158833, A158834.
%K A158832 nonn
%O A158832 1,2
%A A158832 _Paul D. Hanna_, Mar 28 2009