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

1, 1, -4, 33, -376, 5255, -85392, 1566656, -31869104, 710089551, -17178977940, 448256023501, -12548355934560, 375195009917364, -11936772609109600, 402740733371490540, -14367278506882083936, 540452504929440595503, -21384560213508955184172

Offset: 0

Author

Paul D. Hanna, Mar 03 2012

Keywords

• sign

Examples

			G.f.: A(x) = 1 + x - 4*x^2 + 33*x^3 - 376*x^4 + 5255*x^5 - 85392*x^6 +...
where
1/A(x)^2 = 1 - 2*x + 11*x^2 - 94*x^3 + 1051*x^4 - 14232*x^5 +...
The coefficients in A(x)^n begin:
n=1: [1, 1,  -4, 33,  -376,  5255,  -85392, 1566656, ...];
n=2: [1, 2,  -7, 58,  -670,  9494, -156177, 2895672, ...];
n=3: [1, 3,  -9, 76,  -894, 12864, -214339, 4016688, ...];
n=4: [1, 4,(-10),88, -1059, 15496, -261634, 4956000, ...];
n=5: [1, 5,(-10),95, -1175, 17506, -299610, 5736885, ...];
n=6: [1, 6,  -9,(98),-1251, 18996, -329626, 6379902, ...];
n=7: [1, 7,  -7,(98),-1295, 20055, -352870, 6903170, ...];
n=8: [1, 8,  -4, 96,(-1314),20760, -370376, 7322624, ...];
n=9: [1, 9,   0, 93,(-1314),21177, -383040, 7652250, ...];
n=10:[1,10,   5, 90, -1300,(21362),-391635, 7904300, ...];
n=11:[1,11,  11, 88, -1276,(21362),-396825, 8089488, ...];
n=12:[1,12,  18, 88, -1245, 21216,(-399178),8217168, ...];
n=13:[1,13,  26, 91, -1209, 20956,(-399178),8295495, ...];
n=14:[1,14,  35, 98, -1169, 20608, -397236,(8331570), ...];
n=15:[1,15,  45, 110,-1125, 20193, -393700,(8331570), ...]; ...
where the coefficients in parenthesis demonstrate the property:
[x^n] A(x)^(2*n) = [x^n] A(x)^(2*n+1) for n>=2.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..350

Crossrefs

Cf. A185971.

Programs

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)}
for(n=0,25,print1(a(n),", "))

Formula

G.f. A(x) satisfies: [x^n] A(x)^(2*n) = [x^n] A(x)^(2*n+1) for n>=2.

a(n) ~ c * (-1)^(n+1) * n! * 2^n * n^(3/2), where c = 0.18828692660370683384... - Vaclav Kotesovec, Feb 22 2014

Extensions

Typo in name corrected by Vaclav Kotesovec, Feb 22 2014