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A337913
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G.f. A(x) satisfies: Sum_{n>=0} x^n * A(x)^(n*(n-1)/2) = 1 / Sum_{n>=0} (-x)^n * A(x)^(n*(n+1)/2).
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1
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1, 1, 3, 11, 46, 213, 1064, 5617, 30903, 175638, 1025387, 6124272, 37306897, 231245728, 1455929352, 9298583445, 60183716399, 394471770814, 2617028597787, 17567767171467, 119308120317041, 819710477916702, 5698161398891266, 40085629171593620, 285471233080868320, 2058896624654972561
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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 + 3*x^2 + 11*x^3 + 46*x^4 + 213*x^5 + 1064*x^6 + 5617*x^7 + 30903*x^8 + 175638*x^9 + 1025387*x^10 + ...
where the following series illustrate the definition.
B(x) = 1 + x + x^2*A(x) + x^3*A(x)^3 + x^4*A(x)^6 + x^5*A(x)^10 + x^6*A(x)^15 + x^7*A(x)^21 + x^8*A(x)^28 + ... + x^n*A(x)^(n*(n-1)/2) + ...
1/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)^n*A(x)^(n*(n+1)/2) + ...
Explicitly,
B(x) = Sum_{n>=0} x^n * A(x)^(n*(n-1)/2) = 1 + x + x^2 + 2*x^3 + 7*x^4 + 30*x^5 + 142*x^6 + 720*x^7 + 3845*x^8 + 21372*x^9 + 122622*x^10 + 722060*x^11 + 4346304*x^12 + 26665592*x^13 + 166389982*x^14 + 1054232580*x^15 + ...
1/B(x) = Sum_{n>=0} (-x)^n * A(x)^(n*(n+1)/2) = 1 - x - x^3 - 4*x^4 - 18*x^5 - 88*x^6 - 457*x^7 - 2486*x^8 - 14020*x^9 - 81386*x^10 - 483940*x^11 - 2937558*x^12 - 18156724*x^13 - 114054270*x^14 - 727066927*x^15 - ...
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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/sum(m=0, #A, (-x)^m*Ser(A)^(m*(m+1)/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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