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%I #3 Apr 28 2019 20:26:47
%S 1,2,3,6,17,56,189,673,2561,10321,43612,192439,884702,4227202,
%T 20942697,107363291,568547892,3105231155,17467413871,101069173004,
%U 600841031279,3665958252167,22933712331957,146968161483626,963973640814332,6466300466801210,44327544752355141,310325239786656220,2217191324979383686,16157187739844358535,120020165206009363396,908305634422244782653,6999639387956913535113
%N G.f. A(x) satisfies: Sum_{n>=0} x^(n*(n+1)/2) * A(x)^n = Sum_{n>=0} x^n * (1+x)^(n*(n+1)/2).
%e G.f.: A(x) = 1 + 2*x + 3*x^2 + 6*x^3 + 17*x^4 + 56*x^5 + 189*x^6 + 673*x^7 + 2561*x^8 + 10321*x^9 + 43612*x^10 + 192439*x^11 + 884702*x^12 + ...
%e such that the following series are equal
%e B(x) = 1 + x*A(x) + x^3*A(x)^2 + x^6*A(x)^3 + x^10*A(x)^4 + x^15*A(x)^5 + x^21*A(x)^6 + x^28*A(x)^7 + x^36*A(x)^8 + x^45*A(x)^9 + ...
%e B(x) = 1 + x*(1+x) + x^2*(1+x)^3 + x^3*(1+x)^6 + x^4*(1+x)^10 + x^5*(1+x)^15 + x^6*(1+x)^21 + x^7*(1+x)^28 + x^8(1+x)^36 + x^9*(1+x)^45 + ...
%e where
%e B(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 27*x^5 + 81*x^6 + 262*x^7 + 910*x^8 + 3363*x^9 + 13150*x^10 + 54135*x^11 + 233671*x^12 + ... + A121690(n)*x^n + ...
%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*(m+1)/2)*Ser(A)^m - x^m*(1+x +x*O(x^#A) )^(m*(m+1)/2) ),#A) );A[n+1]}
%o for(n=0,35,print1(a(n),", "))
%Y Cf. A121690.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Apr 28 2019
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