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A061261
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Limits of diagonals in triangle defined in A061260.
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2
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1, 2, 6, 15, 37, 85, 194, 423, 912, 1917, 3974, 8096, 16302, 32382, 63668, 123851, 238756, 456190, 864821, 1627016, 3039845, 5641884, 10406924, 19083836, 34802782, 63135539, 113965033, 204739662, 366156396, 652001918, 1156200929, 2042173379, 3593341512
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OFFSET
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0,2
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COMMENTS
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Terms 1, 2, 6, 15, 37, 85, ... are limits of diagonals in the triangle T(n,k) = A061260: 1 2, 1 3, 2, 1, 5, 6, 2, 1, 7, 11, 6, 2, 1, 11, 23, 15, 6, 2, 1, 15, 40, 32, 15, 6, 2, 1, 22, 73, 67, 37, 15, 6, 2, 1, 30, 120, 134, 79, 37, 15, 6, 2, 1, 42, 202, 255, 172, 85, 37, 15, 6, 2, 1, 56, 320, 470, 348, 187, 85, 37, 15, 6, 2, 1
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LINKS
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FORMULA
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G.f.: Product_{k >= 1} (1 - x^k)^( - numbpart(k + 1)), where numbpart(k) = number of partitions of k, cf. A000041. a(n) = 1/n*Sum_{k = 1..n} b(k)*a(n - k), n>0, a(0) = 1, where b(k) = Sum_{d|k} d*numbpart(d + 1).
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MAPLE
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with(numtheory): with(combinat):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
numbpart(d+1), d=divisors(j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..40); # Alois P. Heinz, Apr 14 2017, revised Sep 19 2017
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MATHEMATICA
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a[n_] := a[n] = If[n==0, 1, Sum[Sum[d PartitionsP[d+1], {d, Divisors[j]}] a[n-j], {j, 1, n}]/n];
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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