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A192738
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G.f. satisfies: A(x) = 1/(1 - x*A(x)/(1 - x^2*A(x)/(1 - x^4*A(x)/(1 - x^8*A(x)/(1 - ...))))), a recursive continued fraction.
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2
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1, 1, 2, 6, 19, 64, 225, 817, 3037, 11503, 44237, 172274, 678012, 2692519, 10775673, 43416923, 175973241, 716988597, 2935005104, 12065005233, 49784012457, 206130676553, 856155851503, 3566191166505, 14893485428089, 62350269656535, 261608048386819, 1099926940133938, 4633543246538166, 19554304381037925, 82660879651195338
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) ~ c * d^n / n^(3/2), where d = 4.441030346767249462955690731977198943168711119968067173405434... and c = 0.53428706465230936021267408350544799998698165793457... - Vaclav Kotesovec, Nov 18 2017
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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 19*x^4 + 64*x^5 + 225*x^6 + 817*x^7 + 3037*x^8 + 11503*x^9 + 44237*x^10 + 172274*x^11 + 678012*x^12 + 2692519*x^13 + 10775673*x^14 + 43416923*x^15 + 175973241*x^16 + 716988597*x^17 + 2935005104*x^18 + 12065005233*x^19 + 49784012457*x^20 +...
The g.f. satisfies:
A(x) = 1 + x*A(x) + (x^2 + x^3)*A(x)^2 + (x^3 + 2*x^4 + x^5 + x^7)*A(x)^3 + (x^4 + 3*x^5 + 3*x^6 + x^7 + 2*x^8 + 2*x^9 + x^11 + x^15)*A(x)^4 + (x^5 + 4*x^6 + 6*x^7 + 4*x^8 + 4*x^9 + 6*x^10 + 3*x^11 + 2*x^12 + 3*x^13 + x^15 + 2*x^16 + 2*x^17 + 2*x^19 + x^23 + x^31)*A(x)^5 + (x^6 + 5*x^7 + 10*x^8 + 10*x^9 + 9*x^10 + 13*x^11 + 12*x^12 + 7*x^13 + 9*x^14 + 6*x^15 + 2*x^16 + 7*x^17 + 6*x^18 + 4*x^19 + 4*x^20 + 6*x^21 + 3*x^23 + 2*x^24 + 2*x^25 + 3*x^27 + x^31 + 2*x^32 + 2*x^33 + 2*x^35 + 2*x^39 + x^47 + x^63)*A(x)^6 + (x^7 + 6*x^8 + 15*x^9 + 20*x^10 + 20*x^11 + 26*x^12 + 31*x^13 + 24*x^14 + 23*x^15 + 24*x^16 + 13*x^17 + 16*x^18 + 22*x^19 + 14*x^20 + 15*x^21 + 18*x^22 + 13*x^23 + 6*x^24 + 15*x^25 + 6*x^26 + 7*x^27 + 6*x^28 + 9*x^29 + 6*x^31 + 2*x^32 + 5*x^33 + 6*x^34 + 7*x^35 + 4*x^36 + 6*x^37 + 4*x^39 + 4*x^40 + 4*x^41 + 6*x^43 + 3*x^47 + 2*x^48 + 2*x^49 + 2*x^51 + 3*x^55 + x^63 + 2*x^64 + 2*x^65 + 2*x^67 + 2*x^71 + 2*x^79 + x^95 + x^127)*A(x)^7 +...
which is a series generated by the continued fraction expression.
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PROG
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(PARI) {a(n)=local(A=1+x, CF, M=#binary(n)); for(i=1, n, CF=1+x; for(k=0, M, CF=1/(1-x^(2^(M-k))*A*CF+x*O(x^n))); A=CF); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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