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%I #32 May 29 2018 09:21:15
%S 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,
%T 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,
%U 212,44,13,6,4,2,1,1,0,1,3273,325,56,21,8,6,2,2,1,1,0,1
%N 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.
%C 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
%H Alois P. Heinz, <a href="/A222730/b222730.txt">Rows n = 0..140, flattened</a>
%F Sum_{k=0..n} k*T(n,k) = A066186(n) = n*A000041(n).
%F Sum_{k=1..n} T(n,k) = A014153(n-1) for n>0.
%F Sum_{k=0..n} T(n,k) = n*(n+1)/2*A000041(n) = A000217(n)*A000041(n).
%F (2 * Sum_{k=0..n} T(n,k)) / (Sum_{k=0..n} k*T(n,k)) = n+1 for n>0.
%F T(2*n+1,n+1) = A002865(n).
%e 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].
%e Triangle T(n,k) begins:
%e 0;
%e 0, 1;
%e 3, 2, 1;
%e 11, 6, 0, 1;
%e 36, 10, 3, 0, 1;
%e 79, 21, 3, 1, 0, 1;
%e 186, 33, 7, 3, 1, 0, 1;
%e 345, 59, 9, 4, 1, 1, 0, 1;
%e 672, 89, 20, 4, 4, 1, 1, 0, 1;
%p b:= proc(n, p) option remember; `if`(n=0 and p=0, [1, 0],
%p `if`(p=0, [0$(n+2)], add((l-> subsop(m+2=p*l[1]+l[m+2], l))
%p ([b(n-p*m, p-1)[], 0$(p*m)]), m=0..n/p)))
%p end:
%p T:= n-> subsop(1=NULL, b(n, n))[]:
%p seq(T(n), n=0..14);
%t 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 *)
%Y Columns k=0-10 give: A213679, A103628, A117525, A222731, A222732, A222733, A222734, A222735, A222736, A222737, A222738.
%Y Cf. A000041, A000217, A002865, A014153, A066186.
%K nonn,tabl
%O 0,4
%A _Alois P. Heinz_, Mar 03 2013
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