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A341395
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Coefficient of x^(2*n) in (-1 + Product_{k>=1} (1 + x^k)^k)^n.
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1
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1, 2, 14, 92, 662, 4872, 36578, 278161, 2135902, 16522967, 128574734, 1005321616, 7891885382, 62160038813, 491003317483, 3888045701232, 30854283708670, 245315312649653, 1953735732991919, 15583347966328833, 124463844976490422, 995305632560023009, 7968042676400949882
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) ~ c * d^n / sqrt(n), where d = 8.191928734348241613884260036383361206707761707495484130816183793791732456844... and c = 0.30227512720649344220720362916140286571342247518684432176920275576011986255... - Vaclav Kotesovec, Feb 20 2021
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MAPLE
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g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(d^2/
`if`(d::odd, 1, 2), d=numtheory[divisors](j)), j=1..n)/n)
end:
b:= proc(n, k) option remember; `if`(k<2, g(n+1), (q->
add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
end:
a:= n-> b(n$2):
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MATHEMATICA
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Join[{1}, Table[SeriesCoefficient[(-1 + Product[(1 + x^k)^k, {k, 1, 2 n}])^n, {x, 0, 2 n}], {n, 1, 22}]]
A[n_, k_] := A[n, k] = If[n == 0, 1, k Sum[A[n - j, k] Sum[(-1)^(j/d + 1) d^2, {d, Divisors[j]}], {j, 1, n}]/n]; T[n_, k_] := Sum[(-1)^i Binomial[k, i] A[n, k - i], {i, 0, k}]; Table[T[2 n, n], {n, 0, 22}]
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CROSSREFS
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Cf. A026007, A257675, A270913, A270922, A324595, A341384, A341385, A341386, A341387, A341388, A341390, A341391, A341393, A341394.
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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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