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A222730
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Total sum T(n,k) of parts <= n of multiplicity k in all partitions of n; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
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12
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0, 0, 1, 3, 2, 1, 11, 6, 0, 1, 36, 10, 3, 0, 1, 79, 21, 3, 1, 0, 1, 186, 33, 7, 3, 1, 0, 1, 345, 59, 9, 4, 1, 1, 0, 1, 672, 89, 20, 4, 4, 1, 1, 0, 1, 1163, 145, 22, 11, 4, 2, 1, 1, 0, 1, 2026, 212, 44, 13, 6, 4, 2, 1, 1, 0, 1, 3273, 325, 56, 21, 8, 6, 2, 2, 1, 1, 0, 1
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OFFSET
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0,4
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COMMENTS
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For k > 0, column k is asymptotic to sqrt(3) * (2*k+1) * exp(Pi*sqrt(2*n/3)) / (2 * k^2 * (k+1)^2 * Pi^2) ~ 6 * (2*k+1) * n * p(n) / (k^2 * (k+1)^2 * Pi^2), where p(n) is the partition function A000041(n). - Vaclav Kotesovec, May 29 2018
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LINKS
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FORMULA
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Sum_{k=1..n} T(n,k) = A014153(n-1) for n>0.
(2 * Sum_{k=0..n} T(n,k)) / (Sum_{k=0..n} k*T(n,k)) = n+1 for n>0.
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EXAMPLE
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The partitions of n=4 are [1,1,1,1], [2,1,1], [2,2], [3,1], [4]. Parts <= 4 with multiplicity m=0 sum up to (2+3+4)+(3+4)+(1+3+4)+(2+4)+(1+2+3) = 36, for m=1 the sum is 2+(3+1)+4 = 10, for m=2 the sum is 1+2 = 3, for m=3 the sum is 0, for m=4 the sum is 1 => row 4 = [36, 10, 3, 0, 1].
Triangle T(n,k) begins:
0;
0, 1;
3, 2, 1;
11, 6, 0, 1;
36, 10, 3, 0, 1;
79, 21, 3, 1, 0, 1;
186, 33, 7, 3, 1, 0, 1;
345, 59, 9, 4, 1, 1, 0, 1;
672, 89, 20, 4, 4, 1, 1, 0, 1;
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MAPLE
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b:= proc(n, p) option remember; `if`(n=0 and p=0, [1, 0],
`if`(p=0, [0$(n+2)], add((l-> subsop(m+2=p*l[1]+l[m+2], l))
([b(n-p*m, p-1)[], 0$(p*m)]), m=0..n/p)))
end:
T:= n-> subsop(1=NULL, b(n, n))[]:
seq(T(n), n=0..14);
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MATHEMATICA
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b[n_, p_] := b[n, p] = If[n == 0 && p == 0, {1, 0}, If[p == 0, Array[0&, n+2], Sum[Function[l, ReplacePart[l, m+2 -> p*l[[1]] + l[[m+2]]]][Join[b[n - p*m, p-1] , Array[0&, p*m]]], {m, 0, n/p}]]]; Rest /@ Table[b[n, n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 16 2013, translated from Maple *)
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CROSSREFS
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Columns k=0-10 give: A213679, A103628, A117525, A222731, A222732, A222733, A222734, A222735, A222736, A222737, A222738.
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KEYWORD
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AUTHOR
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STATUS
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approved
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