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A255928
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Expansion of exp( Sum_{n >= 1} A094088(n)*x^n/n ).
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5
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1, 1, 4, 44, 1025, 41693, 2617128, 234091692, 28251572652, 4421489003700, 870650503128708, 210629395976568828, 61405707768736724472, 21231253444779700476672, 8589776776743377081599500, 4020181599664131540547091076, 2155088041310451318611119556661
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OFFSET
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0,3
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COMMENTS
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It appears that this sequence is integer valued.
The o.g.f. A(x) = 1 + x + 4*x^2 + 44*x^3 + ... for this sequence is such that 1 + x*d/dx( log(A(x) ) is the o.g.f. for A094088.
This sequence is the particular case m = 1 of the following general conjecture.
Let m be an integer and consider the sequence u(n) defined by the recurrence u(n) = m*Sum_{k = 0..n-1} binomial(2*n,2*k) *u(k) with the initial condition u(0) = 1. Then the expansion of exp( Sum_{n >= 1} u(n)*x^n/n ) ) has integer coefficients.
Note that u(n), as a polynomial in the variable m, is the n-th row generating polynomial of A241171.
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LINKS
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FORMULA
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O.g.f.: exp(x + 7*x^2/2 + 121*x^3/3 + 3907*x^4/4 + ...) = 1 + x + 4*x^2 + 44*x^3 + 1025*x^4 + ....
a(0) = 1 and a(n) = 1/n*Sum_{k = 0..n-1} A094088(n-k)*a(k) for n >= 1.
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MAPLE
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A094088 := proc (n) option remember; if n = 0 then 1 else add(binomial(2*n, 2*k)*A094088(k), k = 0 .. n-1) end if; end proc:
A255928 := proc (n) option remember; if n = 0 then 1 else add(A094088(n-k)*A255928(k), k = 0 .. n-1)/n end if; end proc:
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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