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A325297
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G.f. A(x) satisfies: Sum_{n>=0} x^n * A(x)^(n*(n+1)/2) = Sum_{n>=0} x^n * (1 + x*A(x))^(n^2).
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1
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1, 1, 2, 6, 23, 106, 552, 3154, 19354, 125787, 857793, 6097668, 44974804, 343043346, 2699235640, 21870283691, 182217929993, 1559505751514, 13698466973187, 123406993956860, 1139530117567175, 10779314404410246, 104402560966604368, 1034827350713543189, 10491687769529216308, 108749411227098836822, 1151846170139701196232, 12460307262968155546661, 137597370654060977361213
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OFFSET
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0,3
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LINKS
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EXAMPLE
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G.f. A(x) = 1 + x + 2*x^2 + 6*x^3 + 23*x^4 + 106*x^5 + 552*x^6 + 3154*x^7 + 19354*x^8 + 125787*x^9 + 857793*x^10 + 6097668*x^11 + 44974804*x^12 + ...
such that the following series are equal:
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^8*A(x)^36 + x^9*A(x)^45 + ...
B(x) = 1 + x*(1+x*A(x)) + x^2*(1+x*A(x))^4 + x^3*(1+x*A(x))^9 + x^4*(1+x*A(x))^16 + x^5*(1+x*A(x))^25 + x^6*(1+x*A(x))^36 + x^7*(1+x*A(x))^49 + ...
where
B(x) = 1 + x + 2*x^2 + 6*x^3 + 22*x^4 + 92*x^5 + 426*x^6 + 2140*x^7 + 11517*x^8 + 65804*x^9 + 396499*x^10 + 2506624*x^11 + 16559723*x^12 + ...
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = -polcoeff( sum(m=0, #A, x^m*( Ser(A)^(m*(m+1)/2) - (1+x*Ser(A))^(m^2) ) ), #A)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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