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A158833 A diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108).

%I A158833 #5 Jul 13 2018 08:19:14
%S A158833 1,3,20,195,2464,38052,693048,14528217,344320262,9100230282,
%T A158833 265305808404,8456446272144,292528760419440,10913859037065560,
%U A158833 436812586581170976,18668379209883807385,848499254768957476312
%N A158833 A diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108).
%C A158833 Triangle A158835 transforms A158832 into this sequence, where A158832 is the previous diagonal in A158825.
%C A158833 Triangle A158835 transforms this sequence into A158834, the next diagonal in A158825.
%e A158833 Array of coefficients in the i-th iteration of x*Catalan(x):
%e A158833 1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,...;
%e A158833 (1),2,6,21,80,322,1348,5814,25674,115566,528528,2449746,...;
%e A158833 1,(3),12,54,260,1310,6824,36478,199094,1105478,6227712,...;
%e A158833 1,4,(20),110,640,3870,24084,153306,993978,6544242,43652340,...;
%e A158833 1,5,30,(195),1330,9380,67844,500619,3755156,28558484,...;
%e A158833 1,6,42,315,(2464),19852,163576,1372196,11682348,100707972,...;
%e A158833 1,7,56,476,4200,(38052),351792,3305484,31478628,303208212,...;
%e A158833 1,8,72,684,6720,67620,(693048),7209036,75915708,807845676,...;
%e A158833 1,9,90,945,10230,113190,1273668,(14528217),167607066,...;
%e A158833 1,10,110,1265,14960,180510,2212188,27454218,(344320262),...;
%e A158833 1,11,132,1650,21164,276562,3666520,49181418,666200106,(9100230282),...; ...
%e A158833 where terms in parenthesis form the initial terms of this sequence.
%t A158833 a[n_] := Module[{x, F, G}, F = InverseSeries[x - x^2 + O[x]^(n+2)]; G = x; For[i = 1, i <= n+1, i++, G = (F /. x -> G)]; Coefficient[G, x, n]];
%t A158833 Array[a, 17] (* _Jean-François Alcover_, Jul 13 2018, from PARI *)
%o A158833 (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 A158833 Cf. A158825, A158831, A158832, A158834.
%K A158833 nonn
%O A158833 1,2
%A A158833 _Paul D. Hanna_, Mar 28 2009