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A222730
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.
12
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
OFFSET
0,4
COMMENTS
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
LINKS
FORMULA
Sum_{k=0..n} k*T(n,k) = A066186(n) = n*A000041(n).
Sum_{k=1..n} T(n,k) = A014153(n-1) for n>0.
Sum_{k=0..n} T(n,k) = n*(n+1)/2*A000041(n) = A000217(n)*A000041(n).
(2 * Sum_{k=0..n} T(n,k)) / (Sum_{k=0..n} k*T(n,k)) = n+1 for n>0.
T(2*n+1,n+1) = A002865(n).
EXAMPLE
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;
MAPLE
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);
MATHEMATICA
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 *)
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Mar 03 2013
STATUS
approved