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 A317801 G.f. A(x) satisfies: Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(2*n) )^n = 1. 6
 1, 2, 5, 36, 465, 8504, 196900, 5448960, 174282930, 6304990774, 254130115519, 11285072891720, 547300883009420, 28782595694322682, 1631647871872693931, 99196878522983084624, 6438935162966517263673, 444501575692314862825620, 32520919561355194120182078, 2513781290973908970634293260, 204713526722520414595009119193 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Paul D. Hanna, Table of n, a(n) for n = 0..200 FORMULA G.f. A(x) satisfies: (1) 1 = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(2*n) )^n. (2) A(x) = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(2*n+2) )^n. (3) 1 = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(2*n+2) )^n / (1+x)^(2*n+2). (4) Let B(x) = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(2*n+1) )^n , then B(x) = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(2*n+2) )^n / (1+x)^(n+1). a(n) ~ 2^(n - log(2)/4 - 5/2) * n^n / (sqrt(1-log(2)) * exp(n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, Aug 13 2018 EXAMPLE G.f.: A(x) = 1 + 2*x + 5*x^2 + 36*x^3 + 465*x^4 + 8504*x^5 + 196900*x^6 + 5448960*x^7 + 174282930*x^8 + 6304990774*x^9 + 254130115519*x^10 + ... such that 1 = 1 + (1/A(x) - 1/(1+x)^2) + (1/A(x) - 1/(1+x)^4)^2 + (1/A(x) - 1/(1+x)^6)^3 + (1/A(x) - 1/(1+x)^8)^4 + (1/A(x) - 1/(1+x)^10)^5 + (1/A(x) - 1/(1+x)^12)^6 + (1/A(x) - 1/(1+x)^14)^7 + (1/A(x) - 1/(1+x)^16)^8 + ... Also, A(x) = 1 + (1/A(x) - 1/(1+x)^4) + (1/A(x) - 1/(1+x)^6)^2 + (1/A(x) - 1/(1+x)^8)^3 + (1/A(x) - 1/(1+x)^10)^4 + (1/A(x) - 1/(1+x)^12)^5 + (1/A(x) - 1/(1+x)^14)^6 + (1/A(x) - 1/(1+x)^16)^7 + (1/A(x) - 1/(1+x)^18)^8 + ... RELATED SERIES. The series B(x) = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(2*n+1) )^n begins B(x) = 1 + x + 2*x^2 + 15*x^3 + 193*x^4 + 3523*x^5 + 81497*x^6 + 2254695*x^7 + 72114516*x^8 + 2609080565*x^9 + 105173092838*x^10 + ... restated, B(x) = 1 + (1/A(x) - 1/(1+x)^3) + (1/A(x) - 1/(1+x)^5)^2 + (1/A(x) - 1/(1+x)^7)^3 + (1/A(x) - 1/(1+x)^9)^4 + (1/A(x) - 1/(1+x)^11)^5 + (1/A(x) - 1/(1+x)^13)^6 + (1/A(x) - 1/(1+x)^15)^7 + (1/A(x) - 1/(1+x)^17)^8 + ... which can also be written B(x) = 1/(1+x) + (1/A(x) - 1/(1+x)^4)/(1+x)^2 + (1/A(x) - 1/(1+x)^6)^2/(1+x)^3 + (1/A(x) - 1/(1+x)^8)^3/(1+x)^4 + (1/A(x) - 1/(1+x)^10)^4/(1+x)^5 + (1/A(x) - 1/(1+x)^12)^5/(1+x)^6 + (1/A(x) - 1/(1+x)^14)^6/(1+x)^7 + (1/A(x) - 1/(1+x)^16)^7/(1+x)^8 + (1/A(x) - 1/(1+x)^18)^8/(1+x)^9 + ... Compare the above to 1 = 1/(1+x)^2 + (1/A(x) - 1/(1+x)^4)/(1+x)^4 + (1/A(x) - 1/(1+x)^6)^2/(1+x)^6 + (1/A(x) - 1/(1+x)^8)^3/(1+x)^8 + (1/A(x) - 1/(1+x)^10)^4/(1+x)^10 + (1/A(x) - 1/(1+x)^12)^5/(1+x)^12 + (1/A(x) - 1/(1+x)^14)^6/(1+x)^14 + (1/A(x) - 1/(1+x)^16)^7/(1+x)^16 + (1/A(x) - 1/(1+x)^18)^8/(1+x)^18 + ... PROG (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ( 1/Ser(A) - 1/(1+x +x*O(x^#A))^(2*m+2) )^m ) )[#A]/2 ); A[n+1]} for(n=0, 25, print1(a(n), ", ")) CROSSREFS Cf. A317339, A317802, A317803, A317666. Sequence in context: A329704 A275552 A086832 * A111491 A331402 A284605 Adjacent sequences: A317798 A317799 A317800 * A317802 A317803 A317804 KEYWORD nonn AUTHOR Paul D. Hanna, Aug 12 2018 STATUS approved

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Last modified March 26 22:49 EDT 2023. Contains 361553 sequences. (Running on oeis4.)