proposed

approved

editing

proposed

nmax = 18;

g[x_] := Module[{y}, Expand[Normal[(1 - Sqrt[1 - 4*y])/2 + O[y]^(nmax+2)] /. y -> x][[1 ;; nmax+1]]];

T = Table[Nest[g, x, n] // CoefficientList[#, x]& // Rest, {n, 1, nmax+1}];

Prepend[Diagonal[T, 1], 1] (* Jean-François Alcover, Jul 13 2018 *)

approved

editing

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

OEIS Server: https://oeis.org/edit/global/213

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

1, 1, 6, 54, 640, 9380, 163576, 3305484, 75915708, 1952409954, 55573310936, 1734182983962, 58863621238500, 2159006675844616, 85088103159523296, 3585740237981536700, 160894462797493581048, 7658326127259130753070

1,3

Triangle A158835 transforms this sequence into A158832, the next diagonal in A158825.

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

(1);

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,(1952409954),...; ...

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, A158832, A158833, A158834.

nonn

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

approved