A193036 G.f. A(x) satisfies: x = Sum_{n>=1} x^n*A(-x)^A006519(n), where A006519(n) is the highest power of 2 dividing n.

1, 1, 1, 3, 10, 34, 112, 382, 1352, 4884, 17856, 66022, 246764, 930878, 3538788, 13542716, 52133416, 201746212, 784378792, 3062431132, 12001867312, 47197716460, 186187480816, 736582735738, 2921679555340, 11617001425938, 46294191726972, 184866924629832

Offset: 0

Author

Paul D. Hanna, Jul 14 2011

Keywords

• nonn

Comments

Compare the g.f. to a g.f. C(x) of the Catalan numbers: x = Sum_{n>=1} x^n*C(-x)^(2*n-1).

Examples

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 10*x^4 + 34*x^5 + 112*x^6 + ...
The g.f. satisfies:
x = x*A(-x) + x^2*A(-x)^2 + x^3*A(-x) + x^4*A(-x)^4 + x^5*A(-x) + x^6*A(-x)^2 + x^7*A(-x) + x^8*A(-x)^8 + x^9*A(-x) + ... + x^n * A(-x)^A006519(n) + ...
where A006519 begins: [1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,16,1,2,1,4,...].
The g.f. also satisfies:
x = x*A(-x)/(1-x^2) + x^2*A(-x)^2/(1-x^4) + x^4*A(-x)^4/(1-x^8) + x^8*A(-x)^8/(1-x^16) + x^16*A(-x)^16/(1-x^32) + x^32*A(-x)^32/(1-x^64) + ...
Related table.
The table of coefficients in A(-x)^(2^n) / (1 - x^(2*2^n)) begins:
n=0: [1, -1, 2, -4, 12, -38, 124, -420, 1476, -5304,  ...];
n=1: [1, -2, 3, -8, 28, -96, 324, -1124, 4024, -14684,  ...];
n=2: [1, -4, 10, -28, 95, -344, 1244, -4512, 16616, -62072, ...];
n=3: [1, -8, 36, -136, 514, -2008, 7924, -31176, 122495, ...];
n=4: [1, -16, 136, -848, 4500, -22032, 103480, -473520, ...];
n=5: [1, -32, 528, -6048, 54632, -418720, 2855088, ...];
n=6: [1, -64, 2080, -45888, 775120, -10720576, 126777952, ...];
n=7: [1, -128, 8256, -358016, 11750304, -311550592, 6955997376, ...];
...
where x = Sum_{n>=0} x^(2^n) * A(-x)^(2^n) / (1 - x^(2*2^n)).
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..720

Crossrefs

Cf. A193037, A193038, A193039, A193040, A006519.

Programs

PARI

{a(n)=local(A=[1]);for(i=1,n,A=concat(A,0);A[#A]=polcoeff(sum(m=1,#A,(-x)^m*Ser(A)^(2^valuation(m,2))),#A));if(n<0,0,A[n+1])}

Formula

G.f. satisfies: x = Sum_{n>=0} x^(2^n) * A(-x)^(2^n) / (1 - x^(2*2^n)).