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

0,4

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).

Links

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

Formula

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

Example

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)).

Prog

(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])}

Crossrefs

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

Sequence in context: A041633 A308499 A034215 * A083580 A255631 A289601

Adjacent sequences: A193033 A193034 A193035 * A193037 A193038 A193039

Keyword

nonn

Author

Paul D. Hanna, Jul 14 2011

Status

approved