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%I #18 Jul 24 2019 20:00:44
%S 1,1,2,7,34,194,1224,8293,59339,443701,3442848,27588356,227529532,
%T 1926646258,16721243096,148562052636,1350062285166,12541860393678,
%U 119063283469925,1154811490210063,11442318733504290,115814699610954732,1197419573869725301,12645763334344402812,136404757181266881472,1502619706308922359493,16901618050427393955502
%N G.f. A(x) satisfies: Sum_{n>=0} A(x)^(n^2+1) * x^n = Sum_{n>=0} (A(x)^n + 1)^n * x^n.
%H Paul D. Hanna, <a href="/A326560/b326560.txt">Table of n, a(n) for n = 0..250</a>
%F G.f. A(x) allows the following sums to be equal:
%F (1) B(x) = Sum_{n>=0} A(x)^(n^2+1) * x^n.
%F (2) B(x) = Sum_{n>=0} (A(x)^n + 1)^n * x^n.
%F (3) B(x) = Sum_{n>=0} A(x)^(n^2) * x^n / (1 - x*A(x)^n)^(n+1).
%e G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 34*x^4 + 194*x^5 + 1224*x^6 + 8293*x^7 + 59339*x^8 + 443701*x^9 + 3442848*x^10 + 27588356*x^11 + 227529532*x^12 + ...
%e such that the following sums are equal
%e B(x) = A(x) + A(x)^2*x + A(x)^5*x^2 + A(x)^10*x^3 + A(x)^17*x^4 + A(x)^26*x^5 + A(x)^37*x^6 + A(x)^50*x^7 + A(x)^65*x^8 + ... + A(x)^(n^2+1)*x^n + ...
%e and
%e B(x) = 1 + (1 + A(x))*x + (1 + A(x)^2)^2*x^2 + (1 + A(x)^3)^3*x^3 + (1 + A(x)^4)^4*x^4 + (1 + A(x)^5)^5*x^5 + ... + (1 + A(x)^n)^n*x^n + ...
%e also
%e B(x) = 1/(1 - x) + A(x)*x/(1 - x*A(x))^2 + A(x)^4*x^2/(1 - x*A(x)^2)^3 + A(x)^9*x^3/(1 - x*A(x)^3)^4 + ... + A(x)^(n^2)*x^n/(1 - x*A(x)^n)^(n+1) + ...
%e where
%e B(x) = 1 + 2*x + 5*x^2 + 18*x^3 + 83*x^4 + 448*x^5 + 2690*x^6 + 17453*x^7 + 120196*x^8 + 868508*x^9 + 6532883*x^10 + 50874263*x^11 + 408560700*x^12 + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = polcoeff( sum(m=0, #A, (Ser(A)^(m) + 1)^m*x^m - Ser(A)^(m^2+1)*x^m ), #A-1)); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A326561, A326562, A326563, A326287, A326275, A325296.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jul 23 2019
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