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A327631
Number T(n,k) of parts in all proper k-times partitions of n; triangle T(n,k), n >= 1, 0 <= k <= n-1, read by rows.
8
1, 1, 2, 1, 5, 3, 1, 11, 21, 12, 1, 19, 61, 74, 30, 1, 34, 205, 461, 432, 144, 1, 53, 474, 1652, 2671, 2030, 588, 1, 85, 1246, 6795, 17487, 23133, 15262, 3984, 1, 127, 2723, 20966, 76264, 148134, 158452, 88194, 19980, 1, 191, 6277, 69812, 360114, 1002835, 1606434, 1483181, 734272, 151080
OFFSET
1,3
COMMENTS
In each step at least one part is replaced by the partition of itself into smaller parts. The parts are not resorted.
T(n,k) is defined for all n>=0 and k>=0. The triangle displays only positive terms. All other terms are zero.
Row n is the inverse binomial transform of the n-th row of array A327618.
LINKS
FORMULA
T(n,k) = Sum_{i=0..k} (-1)^(k-i) * binomial(k,i) * A327618(n,i).
T(n,n-1) = n * A327639(n,n-1) = n * A327643(n) for n >= 1.
EXAMPLE
T(4,0) = 1:
4 (1 part).
T(4,1) = 11 = 2 + 2 + 3 + 4:
4-> 31 (2 parts)
4-> 22 (2 parts)
4-> 211 (3 parts)
4-> 1111 (4 parts)
T(4,2) = 21 = 3 + 4 + 3 + 3 + 4 + 4:
4-> 31 -> 211 (3 parts)
4-> 31 -> 1111 (4 parts)
4-> 22 -> 112 (3 parts)
4-> 22 -> 211 (3 parts)
4-> 22 -> 1111 (4 parts)
4-> 211-> 1111 (4 parts)
T(4,3) = 12 = 4 + 4 + 4:
4-> 31 -> 211 -> 1111 (4 parts)
4-> 22 -> 112 -> 1111 (4 parts)
4-> 22 -> 211 -> 1111 (4 parts)
Triangle T(n,k) begins:
1;
1, 2;
1, 5, 3;
1, 11, 21, 12;
1, 19, 61, 74, 30;
1, 34, 205, 461, 432, 144;
1, 53, 474, 1652, 2671, 2030, 588;
1, 85, 1246, 6795, 17487, 23133, 15262, 3984;
1, 127, 2723, 20966, 76264, 148134, 158452, 88194, 19980;
...
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
`if`(k=0, [1, 1], `if`(i<2, 0, b(n, i-1, k))+
(h-> (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*
b(n-i, min(n-i, i), k)))(b(i$2, k-1))))
end:
T:= (n, k)-> add(b(n$2, i)[2]*(-1)^(k-i)*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n-1), n=1..12);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[k == 0, {1, 1}, If[i < 2, 0, b[n, i - 1, k]] + Function[h, Function[f, f + {0, f[[1]]*h[[2]]/ h[[1]]}][h[[1]]*b[n - i, Min[n - i, i], k]]][b[i, i, k - 1]]]];
T[n_, k_] := Sum[b[n, n, i][[2]]*(-1)^(k - i)*Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 1, 12}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, Jan 07 2020, after Alois P. Heinz *)
CROSSREFS
Columns k=0-2 give: A057427, -1+A006128(n), A328042.
Row sums give A327648.
T(n,floor(n/2)) gives A328041.
Sequence in context: A210792 A105728 A120095 * A130197 A106513 A054446
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 19 2019
STATUS
approved