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 A317802 G.f. A(x) satisfies: Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(3*n) )^n = 1. 6
 1, 3, 12, 127, 2445, 66939, 2324026, 96491718, 4631150520, 251413638241, 15206137508067, 1013223645173301, 73729926406815893, 5817609547850902791, 494790115210979151063, 45129281235546080750387, 4394695321061357601501585, 455127430187799524613334185, 49952816657399856543050669882, 5792366218971732073257841216098, 707622192835283858272032714820854 (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)^(3*n) )^n. (2) A(x) = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(3*n+3) )^n. (3) 1 = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(3*n+3) )^n / (1+x)^(3*n+3). (4) Let B(x) = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(3*n+1) )^n , then B(x) = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(3*n+3) )^n / (1+x)^(2*n+2). (5) Let C(x) = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(3*n+2) )^n , then C(x) = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(3*n+3) )^n / (1+x)^(n+1). a(n) ~ 2^(-log(2)/6 - 5/2) * 3^n * n^n / (sqrt(1-log(2)) * exp(n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, Aug 13 2018 EXAMPLE G.f.: A(x) = 1 + 3*x + 12*x^2 + 127*x^3 + 2445*x^4 + 66939*x^5 + 2324026*x^6 + 96491718*x^7 + 4631150520*x^8 + 251413638241*x^9 + 15206137508067*x^10 + ... such that 1 = 1 + (1/A(x) - 1/(1+x)^3) + (1/A(x) - 1/(1+x)^6)^2 + (1/A(x) - 1/(1+x)^9)^3 + (1/A(x) - 1/(1+x)^12)^4 + (1/A(x) - 1/(1+x)^15)^5 + (1/A(x) - 1/(1+x)^18)^6 + (1/A(x) - 1/(1+x)^21)^7 + (1/A(x) - 1/(1+x)^24)^8 + ... Also, A(x) = 1 + (1/A(x) - 1/(1+x)^6) + (1/A(x) - 1/(1+x)^9)^2 + (1/A(x) - 1/(1+x)^12)^3 + (1/A(x) - 1/(1+x)^15)^4 + (1/A(x) - 1/(1+x)^18)^5 + (1/A(x) - 1/(1+x)^21)^6 + (1/A(x) - 1/(1+x)^24)^7 + (1/A(x) - 1/(1+x)^27)^8 + ... RELATED SERIES. (1) The series B(x) = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(3*n+1) )^n begins B(x) = 1 + x + 3*x^2 + 33*x^3 + 634*x^4 + 17326*x^5 + 601161*x^6 + 24961740*x^7 + 1198455358*x^8 + 65087157334*x^9 + 3938132342935*x^10 + ... restated, B(x) = 1 + (1/A(x) - 1/(1+x)^4) + (1/A(x) - 1/(1+x)^7)^2 + (1/A(x) - 1/(1+x)^10)^3 + (1/A(x) - 1/(1+x)^13)^4 + (1/A(x) - 1/(1+x)^16)^5 + (1/A(x) - 1/(1+x)^19)^6 + (1/A(x) - 1/(1+x)^22)^7 + (1/A(x) - 1/(1+x)^25)^8 + ... which can also be written B(x) = 1/(1+x)^2 + (1/A(x) - 1/(1+x)^6)/(1+x)^4 + (1/A(x) - 1/(1+x)^9)^2/(1+x)^6 + (1/A(x) - 1/(1+x)^12)^3/(1+x)^8 + (1/A(x) - 1/(1+x)^15)^4/(1+x)^10 + (1/A(x) - 1/(1+x)^18)^5/(1+x)^12 + (1/A(x) - 1/(1+x)^21)^6/(1+x)^14 + (1/A(x) - 1/(1+x)^24)^7/(1+x)^16 + (1/A(x) - 1/(1+x)^27)^8/(1+x)^18 + ... ... (2) The series C(x) = Sum_{n>=0} ( 1/A(x) - 1/(1+x)^(3*n+2) )^n begins C(x) = 1 + 2*x + 7*x^2 + 75*x^3 + 1442*x^4 + 39413*x^5 + 1367095*x^6 + 56736076*x^7 + 2722528369*x^8 + 147785496105*x^9 + 8937999326808*x^10 + ... restated, C(x) = 1 + (1/A(x) - 1/(1+x)^5) + (1/A(x) - 1/(1+x)^8)^2 + (1/A(x) - 1/(1+x)^11)^3 + (1/A(x) - 1/(1+x)^14)^4 + (1/A(x) - 1/(1+x)^17)^5 + (1/A(x) - 1/(1+x)^20)^6 + (1/A(x) - 1/(1+x)^23)^7 + (1/A(x) - 1/(1+x)^26)^8 + ... which can also be written C(x) = 1/(1+x) + (1/A(x) - 1/(1+x)^6)/(1+x)^2 + (1/A(x) - 1/(1+x)^9)^2/(1+x)^3 + (1/A(x) - 1/(1+x)^12)^3/(1+x)^4 + (1/A(x) - 1/(1+x)^15)^4/(1+x)^5 + (1/A(x) - 1/(1+x)^18)^5/(1+x)^6 + (1/A(x) - 1/(1+x)^21)^6/(1+x)^7 + (1/A(x) - 1/(1+x)^24)^7/(1+x)^8 + (1/A(x) - 1/(1+x)^27)^8/(1+x)^9 + ... ... Compare the above series to 1 = 1/(1+x)^3 + (1/A(x) - 1/(1+x)^6)/(1+x)^6 + (1/A(x) - 1/(1+x)^9)^2/(1+x)^9 + (1/A(x) - 1/(1+x)^12)^3/(1+x)^12 + (1/A(x) - 1/(1+x)^15)^4/(1+x)^15 + (1/A(x) - 1/(1+x)^18)^5/(1+x)^18 + (1/A(x) - 1/(1+x)^21)^6/(1+x)^21 + (1/A(x) - 1/(1+x)^24)^7/(1+x)^24 + (1/A(x) - 1/(1+x)^27)^8/(1+x)^27 + ... 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))^(3*m+3) )^m ) )[#A]/2 ); A[n+1]} for(n=0, 25, print1(a(n), ", ")) CROSSREFS Cf. A317339, A317801, A317803, A317667. Sequence in context: A124153 A162127 A073987 * A138392 A152544 A280115 Adjacent sequences: A317799 A317800 A317801 * A317803 A317804 A317805 KEYWORD nonn AUTHOR Paul D. Hanna, Aug 12 2018 STATUS approved

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Last modified March 22 01:15 EDT 2023. Contains 361413 sequences. (Running on oeis4.)