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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A041633 A308499 A034215 * A083580 A255631 A289601
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 14 2011
STATUS
approved

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Last modified July 21 05:28 EDT 2024. Contains 374463 sequences. (Running on oeis4.)