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
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
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 *)
Comments