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A326287
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G.f. A(x) satisfies: Sum_{n>=0} A(x)^((n+1)^2) * x^n = Sum_{n>=0} (1 + A(x)^(n+2))^n * x^n.
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7
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1, 1, 2, 9, 53, 357, 2623, 20487, 167571, 1421663, 12430232, 111506176, 1022965373, 9575262402, 91290592979, 885415696549, 8728300482538, 87399936654843, 888643475396035, 9172662449948948, 96116312656950277, 1022513226629210310, 11045623068915556941, 121191713354394285810, 1350988071125342019102, 15306389786282149439384, 176313553377393691773895
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OFFSET
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0,3
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LINKS
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FORMULA
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G.f. A(x) allows the following sums to be equal:
(1) B(x) = Sum_{n>=0} A(x)^((n+1)^2) * x^n.
(2) B(x) = Sum_{n>=0} (A(x)^(n+2) + 1)^n * x^n.
(3) B(x) = Sum_{n>=0} A(x)^(n*(n+2)) * x^n / (1 - x*A(x)^n)^(n+1).
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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 53*x^4 + 357*x^5 + 2623*x^6 + 20487*x^7 + 167571*x^8 + 1421663*x^9 + 12430232*x^10 + ...
such that the following sums are equal:
B(x) = A(x) + A(x)^4*x + A(x)^9*x^2 + A(x)^16*x^3 + A(x)^25*x^4 + A(x)^36*x^5 + A(x)^49*x^6 + A(x)^64*x^7 + ...
and
B(x) = 1 + (1 + A(x)^3)*x + (1 + A(x)^4)^2*x^2 + (1 + A(x)^5)^3*x^3 + (1 + A(x)^6)^4*x^4 + (1 + A(x)^7)^5*x^5 + (1 + A(x)^8)^6*x^6 + (1 + A(x)^9)^7*x^7 + ...
also
B(x) = 1/(1 - x) + A(x)^2*x/(1 - x*A(x))^2 + A(x)^6*x/(1 - x*A(x)^2)^3 + A(x)^12*x/(1 - x*A(x)^3)^4 + A(x)^20*x/(1 - x*A(x)^4)^5 + ...
where
B(x) = 1 + 2*x + 7*x^2 + 33*x^3 + 188*x^4 + 1213*x^5 + 8537*x^6 + 64061*x^7 + 505060*x^8 + 4142471*x^9 + 35103445*x^10 + 305821769*x^11 + 2729269060*x^12 + 24884029770*x^13 + 231326144627*x^14 + 2189345765437*x^15 + 21072597189266*x^16 + 206111127679609*x^17 + 2047583929329250*x^18 + ...
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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, (Ser(A)^(m+2) + 1)^m*x^m - Ser(A)^((m+1)^2)*x^m ), #A-1)); 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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