A158832 - OEIS

%I #7 Jul 13 2018 08:19:45

%S 1,2,12,110,1330,19852,351792,7209036,167607066,4357308098,

%T 125219900520,3941126688798,134808743674176,4979127855477336,

%U 197480359402576304,8370550907396970684,377599345119560766534,18061714498169627460982

%N Main 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 A158831 into this sequence, where A158831 is the previous diagonal in A158825.

%C Triangle A158835 transforms this sequence into A158833, the next diagonal in A158825.

%H Paul D. Hanna, <a href="/A158832/b158832.txt">Table of n, a(n), n = 1..50.</a>

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

%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),...;

%e 1,10,110,1265,14960,180510,2212188,27454218,344320262,(4357308098),...; ...

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

%t 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 Array[a, 18] (* _Jean-François Alcover_, Jul 13 2018, from PARI *)

%o (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 Cf. A158825, A158831, A158833, A158834.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Mar 28 2009