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A293239
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Number of terms in the fully expanded n-th derivative of x^x.
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4
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1, 2, 4, 7, 11, 15, 21, 28, 35, 43, 53, 64, 76, 88, 102, 117, 133, 149, 167, 186, 206, 226, 248, 271, 295, 319, 345, 372, 400, 428, 458, 489, 521, 553, 587, 622, 658, 694, 732, 771, 811, 851, 893, 936, 980, 1024, 1070, 1117, 1165, 1213, 1263, 1314, 1366, 1418
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OFFSET
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0,2
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COMMENTS
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Conjecture: the 2nd differences are eventually periodic: 1, 1, 1, 0, 2, 1, 0, 1, [2, 1, 1, 0].
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LINKS
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FORMULA
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G.f.: (1 + x^2 + x^3 + x^6 - x^8 + x^9 + x^12 - x^13) / ((1 - x)^2*(1 - x^4)).
a(n) = (5 + (-1)^n + (1-i)*(-i)^n + (1+i)*i^n + 2*n + 4*n^2) / 8 for n>7 where i=sqrt(-1).
a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) for n>6.
(End)
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EXAMPLE
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For n = 3, the 3rd derivative of x^x is x^x + 3*x^x*log(x) + 3*x^x*log^2(x) + x^x*log^3(x) + 3*x^(x-1) + 3*x^(x-1)*log(x) - x^(x-2), so a(3) = 7.
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MATHEMATICA
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Join[{1}, Length /@ Rest[NestList[Expand[D[#, x]] &, x^x, 53]]]
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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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