A158834 - OEIS

%I #5 Jul 13 2018 08:18:59

%S 1,4,30,315,4200,67620,1273668,27454218,666200106,17968302638,

%T 533188477536,17261808531552,605452449574320,22870569475477112,

%U 925663441858807096,39964465820186753753,1833332492818402014474

%N A 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 A158833 into this sequence, where A158833 is the previous diagonal in A158825.

%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,...;

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

%e 1,12,156,2106,29120,409682,5841836,84218134,1225314662,(17968302638),...; ...

%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+2, i++, G = (F /. x -> G)]; Coefficient[G, x, n]];

%t Array[a, 17] (* _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+2,G=subst(F,x,G));polcoeff(G,n)}

%Y Cf. A158825, A158831, A158832, A158833.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Mar 28 2009