proposed

approved

editing

proposed

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]];

Array[a, 17] (* Jean-François Alcover, Jul 13 2018, from PARI *)

approved

editing

_Paul D. Hanna (pauldhanna(AT)juno.com), _, Mar 28 2009

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

1, 3, 20, 195, 2464, 38052, 693048, 14528217, 344320262, 9100230282, 265305808404, 8456446272144, 292528760419440, 10913859037065560, 436812586581170976, 18668379209883807385, 848499254768957476312

1,2

Triangle A158835 transforms A158832 into this sequence, where A158832 is the previous diagonal in A158825.

Triangle A158835 transforms this sequence into A158834, the next diagonal in A158825.

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

1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,...;

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

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

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

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

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

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

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

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

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

1,11,132,1650,21164,276562,3666520,49181418,666200106,(9100230282),...; ...

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

(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)}

Cf. A158825, A158831, A158832, A158834.

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Mar 28 2009

approved