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A325286
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G.f. satisfies: 1 = Sum_{n>=0} (1+x)^(n*(n-1)/2) / A(x)^n * 1/2^(n+1).
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3
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1, 1, 3, 25, 343, 6441, 150975, 4203201, 134852079, 4886641681, 197154406591, 8760602600193, 425074860993439, 22363792326962881, 1268239233311498079, 77129745316500047745, 5008173999379887257151, 345838251972031108425345, 25309861534968595801377279, 1956926079593452273940279169, 159406563966400881627947865279, 13645204581985719926987977747329, 1224591755319676016226530026499583, 114980206425267526899287638805977857
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OFFSET
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0,3
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COMMENTS
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a(n) == 1 (mod 2) for n >= 0.
Conjecture: a(n) == 1 (mod 3) iff n is a number whose base-3 representation contains no 2 (cf. A005836), otherwise a(n) == 0 (mod 3).
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LINKS
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FORMULA
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G.f. satisfies:
(1) 1 = Sum_{n>=0} (1+x)^(n*(n-1)/2) / A(x)^n * 1/2^(n+1).
(2) 1 = Sum_{n>=0} (1+x)^(n*(n+1)/2) / A(x)^(n+1) * 1/2^(n+1).
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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2 + 25*x^3 + 343*x^4 + 6441*x^5 + 150975*x^6 + 4203201*x^7 + 134852079*x^8 + 4886641681*x^9 + 197154406591*x^10 + ...
such that
1 = 1/2 + 1/(2^2*A(x)) + (1+x)/(2^3*A(x)^2) + (1+x)^3/(2^4*A(x)^3) + (1+x)^6/(2^5*A(x)^4) + (1+x)^10/(2^6*A(x)^5) + (1+x)^15/(2^7*A(x)^6) + (1+x)^21/(2^8*A(x)^7) + ...
also,
1 = 1/(2*A(x)) + (1+x)/(2*A(x))^2 + (1+x)^3/(2*A(x))^3 + (1+x)^6/(2*A(x))^4 + (1+x)^10/(2*A(x))^5 + (1+x)^15/(2*A(x))^6 + (1+x)^21/(2*A(x))^7 + (1+x)^28/(2*A(x))^8 + ...
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PROG
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(PARI) /* Requires adequate precision */
{a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = round( polcoeff( sum(m=0, 10*#A+100, (1+x+x*O(x^#A))^(m*(m-1)/2)/Ser(A)^m/2^(m+1)*1.), #A-1))); A[n+1]}
for(n=0, 25, 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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