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A185523 a(n) equals the coefficient of x^n in the n-th iteration of x*(1+x)/(1-x) for n>=1.

%I A185523 #12 Jan 06 2025 01:20:40
%S A185523 1,4,30,368,6370,143372,3983518,131891776,5073152322,222405974228,
%T A185523 10948833767454,598115891581232,35907789570992898,2350053890768819484,
%U A185523 166532363675821702206,12703556005092777381120,1037944178579609842602754
%N A185523 a(n) equals the coefficient of x^n in the n-th iteration of x*(1+x)/(1-x) for n>=1.
%H A185523 Paul D. Hanna, <a href="/A185523/b185523.txt">Table of n, a(n) for n = 1..300</a>
%e A185523 Given G(x) = x*(1+x)/(1-x):
%e A185523 G(x) = x + 2*x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 2*x^6 +...
%e A185523 then the initial coefficients of the n-th iterations of G(x) begin:
%e A185523 n=1: [(1), 2, 2, 2, 2, 2, 2, 2, 2, ...];
%e A185523 n=2: [1,(4), 12, 32, 80, 196, 476, 1152, 2784, ...];
%e A185523 n=3: [1, 6,(30), 138, 602, 2542, 10518, 42994, ...];
%e A185523 n=4: [1, 8, 56,(368), 2320, 14216, 85368, 505312, ...];
%e A185523 n=5: [1, 10, 90, 770,(6370), 51450, 408202, 3194978, ...];
%e A185523 n=6: [1, 12, 132, 1392, 14272,(143372), 1418004, 13854368, ...];
%e A185523 n=7: [1, 14, 182, 2282, 27930, 335846,(3983518), 46736466, ...];
%e A185523 n=8: [1, 16, 240, 3488, 49632, 695312, 9623280,(131891776), ...]; ...;
%e A185523 the coefficients in parenthesis form the initial terms of this sequence.
%o A185523 (PARI) {a(n) = my(A=x,G=x*(1+x)/(1-x +x*O(x^n))); for(i=1, n, A=subst(G, x, A+x*O(x^n))); polcoef(A, n)}
%Y A185523 Cf. A185522, A185524.
%K A185523 nonn
%O A185523 1,2
%A A185523 _Paul D. Hanna_, Jan 30 2011