A191769 -id:A191769 - cp's OEIS frontend

A191768 G.f. a(x) satisfies: A(x) = 1 + Sum_{n>=1} x^n*A(x)^A000265(n) where A000265(n) = largest odd divisor of n.

1, 1, 2, 4, 10, 25, 68, 193, 565, 1688, 5136, 15854, 49517, 156191, 496836, 1591924, 5133091, 16643856, 54234349, 177505376, 583272256, 1923482331, 6363842492, 21117432227, 70265970878, 234388421515, 783664894313, 2625748635300

Offset: 0

Paul D. Hanna, Jun 16 2011

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 25*x^5 + 68*x^6 + 193*x^7 +...
The g.f. satisfies the following identities:
A(x) = 1 + x*A(x) + x^2*A(x) + x^3*A(x)^3 + x^4*A(x) + x^5*A(x)^5 + x^6*A(x)^3 + x^7*A(x)^7 + x^8*A(x) +...+ x^n*A(x)^A000265(n) +...
A(x) = 1 + x*A(x)/(1-x^2*A(x)^2) + x^2*A(x)/(1-x^4*A(x)^2) + x^4*A(x)/(1-x^8*A(x)^2) + x^8*A(x)/(1-x^16*A(x)^2) +...

Cf. A191769.

{a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*(A+x*O(x^n))^(m/2^valuation(m,2))));polcoeff(A,n)}

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

A192483 G.f.: A(x) = Sum_{n>=0} x^n * A(x)^A003188(n) where A003188(n) = n XOR floor(n/2).

Original entry on oeis.org

1, 1, 2, 6, 18, 61, 220, 822, 3157, 12378, 49345, 199441, 815467, 3367153, 14020938, 58811032, 248260925, 1053893607, 4496248445, 19268100048, 82902438819, 357987967157, 1550951132419, 6739554074740, 29366902576469, 128287060703669

Offset: 0

Paul D. Hanna, Jul 01 2011

nonn

A003188(n) is the decimal equivalent of the binary Gray code for n; A003188 forms a permutation of the nonnegative integers.

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 18*x^4 + 61*x^5 + 220*x^6 +...
The g.f. A(x) satisfies:
A(x) = 1 + x*A(x) + x^2*A(x)^3 + x^3*A(x)^2 + x^4*A(x)^6 + x^5*A(x)^7 + x^6*A(x)^5 + x^7*A(x)^4 + x^8*A(x)^12 + x^9*A(x)^13 + x^10*A(x)^15 +...
where the powers of A(x) are given by A003188, which begins:
[0,1,3,2,6,7,5,4,12,13,15,14,10,11,9,8,24,25,27,26,30,31,29,...].

Cf. A003188, A191769.

{a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,x^m*(A+x*O(x^n))^bitxor(m,m\2)));polcoeff(A,n)}

cp's OEIS Frontend

A191768 G.f. a(x) satisfies: A(x) = 1 + Sum_{n>=1} x^n*A(x)^A000265(n) where A000265(n) = largest odd divisor of n.

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