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
Alois P. Heinz, Rows n = 0..140, flattened
FORMULA
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 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Mar 03 2013
STATUS
approved