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A208961 G.f. satisfies: A(x) = 1 + x*[d/dx x/A(x)^2].

%I A208961 #11 Jul 24 2025 19:38:41
%S A208961 1,1,-4,33,-376,5255,-85392,1566656,-31869104,710089551,-17178977940,
%T A208961 448256023501,-12548355934560,375195009917364,-11936772609109600,
%U A208961 402740733371490540,-14367278506882083936,540452504929440595503,-21384560213508955184172
%N A208961 G.f. satisfies: A(x) = 1 + x*[d/dx x/A(x)^2].
%H A208961 Vaclav Kotesovec, <a href="/A208961/b208961.txt">Table of n, a(n) for n = 0..350</a>
%F A208961 G.f. A(x) satisfies: [x^n] A(x)^(2*n) = [x^n] A(x)^(2*n+1) for n>=2.
%F A208961 a(n) ~ c * (-1)^(n+1) * n! * 2^n * n^(3/2), where c = 0.18828692660370683384... - _Vaclav Kotesovec_, Feb 22 2014
%e A208961 G.f.: A(x) = 1 + x - 4*x^2 + 33*x^3 - 376*x^4 + 5255*x^5 - 85392*x^6 +...
%e A208961 where
%e A208961 1/A(x)^2 = 1 - 2*x + 11*x^2 - 94*x^3 + 1051*x^4 - 14232*x^5 +...
%e A208961 The coefficients in A(x)^n begin:
%e A208961 n=1: [1, 1,  -4, 33,  -376,  5255,  -85392, 1566656, ...];
%e A208961 n=2: [1, 2,  -7, 58,  -670,  9494, -156177, 2895672, ...];
%e A208961 n=3: [1, 3,  -9, 76,  -894, 12864, -214339, 4016688, ...];
%e A208961 n=4: [1, 4,(-10),88, -1059, 15496, -261634, 4956000, ...];
%e A208961 n=5: [1, 5,(-10),95, -1175, 17506, -299610, 5736885, ...];
%e A208961 n=6: [1, 6,  -9,(98),-1251, 18996, -329626, 6379902, ...];
%e A208961 n=7: [1, 7,  -7,(98),-1295, 20055, -352870, 6903170, ...];
%e A208961 n=8: [1, 8,  -4, 96,(-1314),20760, -370376, 7322624, ...];
%e A208961 n=9: [1, 9,   0, 93,(-1314),21177, -383040, 7652250, ...];
%e A208961 n=10:[1,10,   5, 90, -1300,(21362),-391635, 7904300, ...];
%e A208961 n=11:[1,11,  11, 88, -1276,(21362),-396825, 8089488, ...];
%e A208961 n=12:[1,12,  18, 88, -1245, 21216,(-399178),8217168, ...];
%e A208961 n=13:[1,13,  26, 91, -1209, 20956,(-399178),8295495, ...];
%e A208961 n=14:[1,14,  35, 98, -1169, 20608, -397236,(8331570), ...];
%e A208961 n=15:[1,15,  45, 110,-1125, 20193, -393700,(8331570), ...]; ...
%e A208961 where the coefficients in parenthesis demonstrate the property:
%e A208961 [x^n] A(x)^(2*n) = [x^n] A(x)^(2*n+1) for n>=2.
%o A208961 (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*deriv(x/A^2));polcoeff(A,n)}
%o A208961 for(n=0,25,print1(a(n),", "))
%Y A208961 Cf. A185971.
%K A208961 sign
%O A208961 0,3
%A A208961 _Paul D. Hanna_, Mar 03 2012
%E A208961 Typo in name corrected by _Vaclav Kotesovec_, Feb 22 2014