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.

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

Author

Alois P. Heinz, Mar 03 2013

Keywords

• nonn
• tabl

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

Examples

			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;
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Columns k=0-10 give: A213679, A103628, A117525, A222731, A222732, A222733, A222734, A222735, A222736, A222737, A222738.

Cf. A000041, A000217, A002865, A014153, A066186.

Programs

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 *)

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).