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A308543
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Expansion of e.g.f. exp(2*(exp(2*x) - 1)).
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2
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1, 4, 24, 176, 1504, 14528, 155520, 1819392, 23019008, 312413184, 4518705152, 69279690752, 1120856170496, 19062628335616, 339681346551808, 6323658075340800, 122680376836358144, 2474677219852288000, 51799971194270646272, 1123121391647711035392
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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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O.g.f.: Sum_{k>=0} 4^k*x^k / Product_{j=1..k} (1 - 2*j*x).
E.g.f.: exp(4*exp(x)*sinh(x)).
E.g.f.: g(g(x) - 1), where g(x) = e.g.f. of A000079 (powers of 2).
E.g.f.: f(x)^4, where f(x) = e.g.f. of A004211 (shifts one place left under 2nd-order binomial transform).
a(0) = 1; a(n) = Sum_{k=1..n} 2^(k+1)*binomial(n-1,k-1)*a(n-k).
a(n) = Sum_{k=0..n} 2^(n+k)*Stirling2(n,k).
a(n) = exp(-2) * Sum_{k>=0} 2^(n+k)*k^n/k!.
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MATHEMATICA
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nmax = 19; CoefficientList[Series[Exp[2 (Exp[2 x] - 1)], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[Sum[4^k x^k/Product[(1 - 2 j x), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = Sum[2^(k + 1) Binomial[n - 1, k - 1] a[n - k], {k, n}]; a[0] = 1; Table[a[n], {n, 0, 19}]
Table[2^n BellB[n, 2], {n, 0, 19}]
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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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