This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A317667 #17 Aug 13 2018 03:54:52
%S A317667 1,3,15,154,2865,77532,2684504,111490839,5357828286,291299582266,
%T A317667 17643988446921,1177175235308976,85754781272021397,
%U A317667 6772714984220704506,576470959628636447748,52613628461306161087953,5126338275850981999654524,531146069930403178373329794,58319563977901655667747310206,6764879932357508722274792757285
%N A317667 G.f. A(x) satisfies: Sum_{n>=0} ( 1/A(x) - (1-x)^(3*n) )^n = 1.
%H A317667 Paul D. Hanna, <a href="/A317667/b317667.txt">Table of n, a(n) for n = 0..200</a>
%F A317667 G.f. A(x) satisfies:
%F A317667 (1) 1 = Sum_{n>=0} ( 1/A(x) - (1-x)^(3*n) )^n.
%F A317667 (2) A(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(3*n+3) )^n.
%F A317667 (3) 1 = Sum_{n>=0} ( 1/A(x) - (1-x)^(3*n+3) )^n * (1-x)^(3*n+3).
%F A317667 (4) Let B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(3*n+1) )^n,
%F A317667 then B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(3*n+3) )^n * (1-x)^(2*n+2).
%F A317667 (5) Let C(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(3*n+2) )^n,
%F A317667 then C(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(3*n+3) )^n * (1-x)^(n+1).
%F A317667 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
%e A317667 G.f.: A(x) = 1 + 3*x + 15*x^2 + 154*x^3 + 2865*x^4 + 77532*x^5 + 2684504*x^6 + 111490839*x^7 + 5357828286*x^8 + 291299582266*x^9 + 17643988446921*x^10 + ...
%e A317667 such that
%e A317667 1 = 1 + (1/A(x) - (1-x)^3) + (1/A(x) - (1-x)^6)^2 + (1/A(x) - (1-x)^9)^3 + (1/A(x) - (1-x)^12)^4 + (1/A(x) - (1-x)^15)^5 + (1/A(x) - (1-x)^18)^6 + (1/A(x) - (1-x)^21)^7 + (1/A(x) - (1-x)^24)^8 + ...
%e A317667 Also,
%e A317667 A(x) = 1 + (1/A(x) - (1-x)^6) + (1/A(x) - (1-x)^9)^2 + (1/A(x) - (1-x)^12)^3 + (1/A(x) - (1-x)^15)^4 + (1/A(x) - (1-x)^18)^5 + (1/A(x) - (1-x)^21)^6 + (1/A(x) - (1-x)^24)^7 + (1/A(x) - (1-x)^27)^8 + ...
%e A317667 RELATED SERIES.
%e A317667 (1) The series B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(3*n+1) )^n begins
%e A317667 B(x) = 1 + x + 4*x^2 + 40*x^3 + 743*x^4 + 20073*x^5 + 694477*x^6 + 28841790*x^7 + 1386441234*x^8 + 75408643207*x^9 + 4569235921823*x^10 + ...
%e A317667 restated,
%e A317667 B(x) = 1 + (1/A(x) - (1-x)^4) + (1/A(x) - (1-x)^7)^2 + (1/A(x) - (1-x)^10)^3 + (1/A(x) - (1-x)^13)^4 + (1/A(x) - (1-x)^16)^5 + (1/A(x) - (1-x)^19)^6 + (1/A(x) - (1-x)^22)^7 + (1/A(x) - (1-x)^25)^8 + ...
%e A317667 which can also be written
%e A317667 B(x) = (1-x)^2 + (1/A(x) - (1-x)^6)*(1-x)^4 + (1/A(x) - (1-x)^9)^2*(1-x)^6 + (1/A(x) - (1-x)^12)^3*(1-x)^8 + (1/A(x) - (1-x)^15)^4*(1-x)^10 + (1/A(x) - (1-x)^18)^5*(1-x)^12 + (1/A(x) - (1-x)^21)^6*(1-x)^14 + (1/A(x) - (1-x)^24)^7*(1-x)^16 + (1/A(x) - (1-x)^27)^8*(1-x)^18 + ...
%e A317667 ...
%e A317667 (2) The series C(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(3*n+2) )^n begins
%e A317667 C(x) = 1 + 2*x + 9*x^2 + 91*x^3 + 1690*x^4 + 45661*x^5 + 1579367*x^6 + 65559850*x^7 + 3149821447*x^8 + 171233732325*x^9 + 10371022987322*x^10 + ...
%e A317667 restated,
%e A317667 C(x) = 1 + (1/A(x) - (1-x)^5) + (1/A(x) - (1-x)^8)^2 + (1/A(x) - (1-x)^11)^3 + (1/A(x) - (1-x)^14)^4 + (1/A(x) - (1-x)^17)^5 + (1/A(x) - (1-x)^20)^6 + (1/A(x) - (1-x)^23)^7 + (1/A(x) - (1-x)^26)^8 + ...
%e A317667 which can also be written
%e A317667 C(x) = (1-x) + (1/A(x) - (1-x)^6)*(1-x)^2 + (1/A(x) - (1-x)^9)^2*(1-x)^3 + (1/A(x) - (1-x)^12)^3*(1-x)^4 + (1/A(x) - (1-x)^15)^4*(1-x)^5 + (1/A(x) - (1-x)^18)^5*(1-x)^6 + (1/A(x) - (1-x)^21)^6*(1-x)^7 + (1/A(x) - (1-x)^24)^7*(1-x)^8 + (1/A(x) - (1-x)^27)^8*(1-x)^9 + ...
%e A317667 ...
%e A317667 Compare the above series to
%e A317667 1 = (1-x)^3 + (1/A(x) - (1-x)^6)*(1-x)^6 + (1/A(x) - (1-x)^9)^2*(1-x)^9 + (1/A(x) - (1-x)^12)^3*(1-x)^12 + (1/A(x) - (1-x)^15)^4*(1-x)^15 + (1/A(x) - (1-x)^18)^5*(1-x)^18 + (1/A(x) - (1-x)^21)^6*(1-x)^21 + (1/A(x) - (1-x)^24)^7*(1-x)^24 + (1/A(x) - (1-x)^27)^8*(1-x)^27 + ...
%o A317667 (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-x)^(3*m+3) )^m ) )[#A]/2 ); A[n+1]}
%o A317667 for(n=0, 25, print1(a(n), ", "))
%Y A317667 Cf. A317349, A317666, A317668, A317802.
%K A317667 nonn
%O A317667 0,2
%A A317667 _Paul D. Hanna_, Aug 12 2018