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A300353
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Number of strict trees of weight n with odd leaves.
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7
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1, 1, 0, 1, 1, 2, 2, 4, 7, 14, 24, 46, 92, 186, 368, 750, 1529, 3160, 6510, 13590, 28374, 59780, 125732, 266468, 564188, 1202842, 2560106, 5484304, 11732400, 25229068, 54187918, 116938702, 252039411, 545593378, 1179545874, 2560009400, 5550315640, 12075064432
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OFFSET
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0,6
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COMMENTS
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This sequence is initially dominated by A300352 but eventually becomes much greater.
A strict tree of weight n > 0 is either a single node of weight n, or a sequence of two or more strict trees with strictly decreasing weights summing to n.
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LINKS
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FORMULA
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O.g.f: (1 + x/(1-x^2) + Product_{i>0} (1 + a(i)x^i))/2.
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EXAMPLE
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The a(8) = 7 strict trees with odd leaves: (71), (53), (((51)1)1), (((31)3)1), (((31)1)3), ((31)31), ((((31)1)1)1)1).
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MATHEMATICA
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d[n_]:=d[n]=If[EvenQ[n], 0, 1]+Sum[Times@@d/@y, {y, Select[IntegerPartitions[n], Length[#]>1&&UnsameQ@@#&]}];
Table[d[n], {n, 40}]
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PROG
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(PARI) seq(n)={my(v=vector(n)); v[1]=1; for(n=2, n, v[n] = polcoef(x/(1-x^2) + prod(k=1, n-1, 1 + v[k]*x^k + O(x*x^n)), n)); concat([1], v)} \\ Andrew Howroyd, Aug 25 2018
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CROSSREFS
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Cf. A000009, A063834, A078408, A089259, A196545, A279374, A279785, A289501, A294018, A294079, A299203, A300300, A300301, A300351, A300352, A300354, A300355.
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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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