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%I #3 Jan 24 2018 23:03:39
%S 1,1,2,10,83,971,14679,271065,5887674,146573343,4106195739,
%T 127709962780,4364136955874,162503129082497,6548680061635319,
%U 283973223632787150,13185195626147207058,652695122347799336199,34316223642036784123819,1909798106976656110119169,112165977515060359849066878,6933265352057611483132200642
%N G.f. A(x) satisfies: A(x) = Sum_{n>=0} binomial( n*(n+1)/2, n) * x^n / A(x)^( n*(n+1)/2 ).
%e G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 83*x^4 + 971*x^5 + 14679*x^6 + 271065*x^7 + 5887674*x^8 + 146573343*x^9 + 4106195739*x^10 + 127709962780*x^11 + 4364136955874*x^12 + 162503129082497*x^13 + 6548680061635319*x^14 + 283973223632787150*x^15 + ...
%e such that
%e A(x) = 1 + C(1,1)*x/A(x) + C(3,2)*x^2/A(x)^3 + C(6,3)*x^3/A(x)^6 + C(10,4)*x^4/A(x)^10 + C(15,5)*x^5/A(x)^15 + C(21,6)*x^6/A(x)^21 + C(28,7)*x^7/A(x)^28 + ...
%e more explicitly,
%e A(x) = 1 + x/A(x) + 3*x^2/A(x)^3 + 20*x^3/A(x)^6 + 210*x^4/A(x)^10 + 3003*x^5/A(x)^15 + 54264*x^6/A(x)^21 + 1184040*x^7/A(x)^28 + 30260340*x^8/A(x)^36 + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A = Vec(sum(m=0,#A,binomial(m*(m+1)/2,m) * x^m/Ser(A)^(m*(m+1)/2) ))); G=Ser(A); A[n+1]}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A014068, A298689, A298691.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 24 2018
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