Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.
%I #7 Oct 02 2020 04:03:46
%S 1,1,3,11,46,213,1064,5617,30903,175638,1025387,6124272,37306897,
%T 231245728,1455929352,9298583445,60183716399,394471770814,
%U 2617028597787,17567767171467,119308120317041,819710477916702,5698161398891266,40085629171593620,285471233080868320,2058896624654972561
%N G.f. A(x) satisfies: Sum_{n>=0} x^n * A(x)^(n*(n-1)/2) = 1 / Sum_{n>=0} (-x)^n * A(x)^(n*(n+1)/2).
%H Paul D. Hanna, <a href="/A337913/b337913.txt">Table of n, a(n) for n = 0..200</a>
%e G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 46*x^4 + 213*x^5 + 1064*x^6 + 5617*x^7 + 30903*x^8 + 175638*x^9 + 1025387*x^10 + ...
%e where the following series illustrate the definition.
%e B(x) = 1 + x + x^2*A(x) + x^3*A(x)^3 + x^4*A(x)^6 + x^5*A(x)^10 + x^6*A(x)^15 + x^7*A(x)^21 + x^8*A(x)^28 + ... + x^n*A(x)^(n*(n-1)/2) + ...
%e 1/B(x) = 1 - x*A(x) + x^2*A(x)^3 - x^3*A(x)^6 + x^4*A(x)^10 - x^5*A(x)^15 + x^6*A(x)^21 - x^7*A(x)^28 +- ... + (-x)^n*A(x)^(n*(n+1)/2) + ...
%e Explicitly,
%e B(x) = Sum_{n>=0} x^n * A(x)^(n*(n-1)/2) = 1 + x + x^2 + 2*x^3 + 7*x^4 + 30*x^5 + 142*x^6 + 720*x^7 + 3845*x^8 + 21372*x^9 + 122622*x^10 + 722060*x^11 + 4346304*x^12 + 26665592*x^13 + 166389982*x^14 + 1054232580*x^15 + ...
%e 1/B(x) = Sum_{n>=0} (-x)^n * A(x)^(n*(n+1)/2) = 1 - x - x^3 - 4*x^4 - 18*x^5 - 88*x^6 - 457*x^7 - 2486*x^8 - 14020*x^9 - 81386*x^10 - 483940*x^11 - 2937558*x^12 - 18156724*x^13 - 114054270*x^14 - 727066927*x^15 - ...
%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
%o A[#A] = polcoeff( sum(m=0,#A, x^m*Ser(A)^(m*(m-1)/2)) - 1/sum(m=0,#A, (-x)^m*Ser(A)^(m*(m+1)/2) ),#A););A[n+1]}
%o for(n=0,30, print1(a(n),", "))
%K nonn
%O 0,3
%A _Paul D. Hanna_, Sep 29 2020