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 A224884 #44 Jun 20 2025 08:11:28
%S A224884 1,2,-6,32,-210,1536,-12012,98304,-831402,7208960,-63740820,572522496,
%T A224884 -5209363380,47915728896,-444799488600,4161823309824,-39209074920090,
%U A224884 371626340253696,-3541117629057540,33902753847705600,-325969196485349340,3146175557067079680,-30471769822097981160
%N A224884 Expansion of x / Series_Reversion(x*sqrt(1 + 4*x)).
%C A224884 Signed version of A206300. - _Peter Bala_, Mar 05 2020
%H A224884 G. C. Greubel, <a href="/A224884/b224884.txt">Table of n, a(n) for n = 0..980</a>
%F A224884 G.f. A(x) satisfies:
%F A224884 (1) A(x) = A(x)^3 - 4*x.
%F A224884 (2) A(x) = sqrt(1 + 4*x/A(x)).
%F A224884 (3) A(x*sqrt(1+4*x)) = sqrt(1+4*x).
%F A224884 (4) [x^n] A(x)^(n+2*k) = 0 for k=1..n-1, for n >= 2.
%F A224884 From _Vaclav Kotesovec_, Aug 22 2013: (Start)
%F A224884 a(n) = (-1)^(n+1) * 3^(3*n/2-1) * 4^(n-1) * GAMMA(n/2 - 1/6) * GAMMA(n/2 + 1/6)/(Pi*n!).
%F A224884 |a(n)| ~ 6^(n-1)*3^(n/2)/(sqrt(Pi/2)*n^(3/2)).
%F A224884 D-finite with recurrence: (n-1)*n*a(n) = 12*(3*n-7)*(3*n-5)*a(n-2). (End)
%F A224884 G.f.: (2/sqrt(3))*cosh(1/3*arccosh(sqrt(108)*x)). - _Vladimir Kruchinin_, Oct 11 2022
%F A224884 G.f. A(x) satisfies A(x) = 1/A(-x/A(x)^4). - _Seiichi Manyama_, Jun 20 2025
%e A224884 G.f.: A(x) = 1 + 2*x - 6*x^2 + 32*x^3 - 210*x^4 + 1536*x^5 - 12012*x^6 + ..
%e A224884 The coefficients in the powers A(x)^n of the g.f. begin:
%e A224884 n= 1: [1, 2, -6, 32, -210, 1536,-12012, 98304, -831402, ...];
%e A224884 n= 2: [1, 4, -8, 40, -256, 1848,-14336, 116688, -983040, ...];
%e A224884 n= 3: [1, 6, -6, 32, -210, 1536,-12012, 98304, -831402, ...];
%e A224884 n= 4: [1, 8, 0, 16, -128, 1008, -8192, 68640, -589824, ...];
%e A224884 n= 5: [1, 10, 10, 0, -50, 512, -4620, 40960, -364650, ...];
%e A224884 n= 6: [1, 12, 24, -8, 0, 168, -2048, 20592, -196608, ...];
%e A224884 n= 7: [1, 14, 42, 0, 14, 0, -588, 8192, -90090, ...];
%e A224884 n= 8: [1, 16, 64, 32, 0, -32, 0, 2112, -32768, ...];
%e A224884 n= 9: [1, 18, 90, 96, -18, 0, 84, 0, -7722, ...];
%e A224884 n=10: [1, 20, 120, 200, 0, 24, 0, -240, 0, ...];
%e A224884 n=11: [1, 22, 154, 352, 110, 0, -44, 0, 726, ...];
%e A224884 n=12: [1, 24, 192, 560, 384, -48, 0, 96, 0, ...];
%e A224884 n=13: [1, 26, 234, 832, 910, 0, 52, 0, -234, ...];
%e A224884 n=14: [1, 28, 280, 1176, 1792, 392, 0, -80, 0, ...];
%e A224884 n=15: [1, 30, 330, 1600, 3150, 1536, -140, 0, 150, ...];
%e A224884 n=16: [1, 32, 384, 2112, 5120, 4032, 0, 128, 0, ...];
%e A224884 n=17: [1, 34, 442, 2720, 7854, 8704, 1428, 0, -170, ...];
%e A224884 n=18: [1, 36, 504, 3432, 11520, 16632, 6144, -432, 0, ...];
%e A224884 n=19: [1, 38, 570, 4256, 16302, 29184, 17556, 0, 342, ...];
%e A224884 n=20: [1, 40, 640, 5200, 22400, 48048, 40960, 5280, 0, ...]; ...
%e A224884 which illustrates the property [x^n] A(x)^(n+2*k) = 0 for k=1..n-1:
%e A224884 [x^2] A(x)^4 = 0;
%e A224884 [x^3] A(x)^5 = 0, [x^3] A(x)^7 = 0;
%e A224884 [x^4] A(x)^6 = 0, [x^4] A(x)^8 = 0, [x^4] A(x)^10 = 0; ...
%e A224884 [x^5] A(x)^7 = 0, [x^5] A(x)^9 = 0, [x^5] A(x)^11 = 0, [x^5] A(x)^13 = 0; ...
%e A224884 Related series:
%e A224884 sqrt(1+4*x) = 1 + 2*x - 2*x^2 + 4*x^3 - 10*x^4 + 28*x^5 - 84*x^6 + 264*x^7 - 858*x^8 + ... + (-1)^(n-1)*2*A000108(n-1)*x^n + ...
%t A224884 CoefficientList[Series[x/InverseSeries[Series[x*Sqrt[1+4*x],{x,0,20}],x],{x,0,20}],x] (* _Vaclav Kotesovec_, Aug 22 2013 *)
%o A224884 (PARI) {a(n)=polcoeff(x/serreverse(x*sqrt(1+4*x +x^2*O(x^n))),n)}
%o A224884 for(n=0,25,print1(a(n),", "))
%Y A224884 Cf. A000108, A206300.
%Y A224884 Cf. A000984, A048990, A078531, A182122, A245112, A245113.
%K A224884 sign
%O A224884 0,2
%A A224884 _Paul D. Hanna_, Aug 21 2013