This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A192947 #11 Feb 01 2017 11:55:07
%S A192947 1,1,4,38,444,5805,81284,1192144,18078660,281172017,4460264072,
%T A192947 71886775636,1173832034804,19377733213699,322866234066016,
%U A192947 5422493523853024,91701823351874276,1560232214582865621,26688686144512908492
%N A192947 G.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^4 - 1)^n.
%C A192947 Compare to a g.f. of the Catalan numbers: C(x) = 1 + x*Sum_{n>=0} (C(x) - 1)^n.
%H A192947 G. C. Greubel, <a href="/A192947/b192947.txt">Table of n, a(n) for n = 0..785</a>
%F A192947 G.f. A(x) equals the formal inverse of function (x-1)*(2-x^4).
%F A192947 G.f. satisfies: A(x) = 1 + x/(2 - A(x)^4).
%F A192947 G.f.: A(x) = 1 + Series_Reversion( 2*x - x*(1+x)^4 ).
%F A192947 Recurrence: 2048*(n-3)*(n-2)*(n-1)*n*(150965*n^3 - 1545075*n^2 + 5236171*n - 5868843)*a(n) = 256*(n-3)*(n-2)*(n-1)*(21135100*n^4 - 248013150*n^3 + 1057496990*n^2 - 1920941631*n + 1231831665)*a(n-1) + 32*(n-3)*(n-2)*(204104680*n^5 - 2905360120*n^4 + 16233323207*n^3 - 44427578447*n^2 + 59458979970*n - 31076616060)*a(n-2) + 8*(n-3)*(372581620*n^6 - 6607607250*n^5 + 48312584548*n^4 - 186256666080*n^3 + 398929297639*n^2 - 449551482162*n + 207932551470)*a(n-3) + 5*(5*n - 21)*(5*n - 19)*(5*n - 18)*(5*n - 17)*(150965*n^3 - 1092180*n^2 + 2598916*n - 2026782)*a(n-4). - _Vaclav Kotesovec_, Sep 17 2013
%F A192947 a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 35/8 + 1/(8*sqrt(3/(4351 + (6495403 + 452895*sqrt(831))^(1/3)/2^(2/3) - 40027/(12990806 + 905790*sqrt(831))^(1/3)))) + 1/2*sqrt(4351/24 - (6495403 + 452895*sqrt(831))^(1/3)/(48*2^(2/3)) + 40027/(48*(12990806 + 905790*sqrt(831))^(1/3)) + 27661/4*sqrt(3/(4351 + (6495403 + 452895*sqrt(831))^(1/3)/2^(2/3) - 40027/(12990806 + 905790*sqrt(831))^(1/3)))) = 18.6600216048327281... is the root of the equation -3125 - 19744*d - 43264*d^2 - 35840*d^3 + 2048*d^4 = 0 and c = 0.047032341973499367520535028629... - _Vaclav Kotesovec_, Sep 17 2013
%e A192947 G.f.: A(x) = 1 + x + 4*x^2 + 38*x^3 + 444*x^4 + 5805*x^5 +...
%e A192947 where (A(x) - 1)*(2 - A(x)^4) = x
%e A192947 and A(x - 4*x^2 - 6*x^3 - 4*x^4 - x^5) = 1 + x.
%e A192947 Related expansions.
%e A192947 (A(x)^4-1) = 4*x + 22*x^2 + 204*x^3 + 2377*x^4 + 31036*x^5 +...
%e A192947 (A(x)^4-1)^2 = 16*x^2 + 176*x^3 + 2116*x^4 + 27992*x^5 +...
%e A192947 (A(x)^4-1)^3 = 64*x^3 + 1056*x^4 + 15600*x^5 + 232456*x^6 +...
%e A192947 (A(x)^4-1)^4 = 256*x^4 + 5632*x^5 + 98688*x^6 + 1640576*x^7 +...
%t A192947 CoefficientList[1+InverseSeries[Series[2*x-x*(1+x)^4,{x,0,20}],x],x] (* _Vaclav Kotesovec_, Sep 17 2013 *)
%o A192947 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+x*sum(m=0,n,(A^4-1+x*O(x^n))^m));polcoeff(A,n)}
%o A192947 (PARI) {a(n)=local(A=1+serreverse(2*x-x*(1+x)^4+x^2*O(x^n)));polcoeff(A,n)}
%Y A192947 Cf. A192945, A192946, A192948.
%K A192947 nonn
%O A192947 0,3
%A A192947 _Paul D. Hanna_, Jul 13 2011