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A256130
Number T(n,k) of partitions of n into parts of exactly k sorts which are introduced in ascending order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
17
1, 0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 5, 12, 7, 1, 0, 7, 30, 33, 11, 1, 0, 11, 72, 130, 77, 16, 1, 0, 15, 160, 463, 438, 157, 22, 1, 0, 22, 351, 1557, 2216, 1223, 289, 29, 1, 0, 30, 743, 5031, 10422, 8331, 2957, 492, 37, 1, 0, 42, 1561, 15877, 46731, 52078, 26073, 6401, 788, 46, 1
OFFSET
0,5
COMMENTS
In general, column k>1 is asymptotic to c*k^n, where c = 1/(k!*Product_{n>=1} (1-1/k^n)) = 1/(k!*QPochhammer[1/k, 1/k]). - Vaclav Kotesovec, Jun 01 2015
LINKS
FORMULA
T(n,k) = A255970(n,k)/k! = (Sum_{i=0..k} (-1)^i * C(k,i) * A246935(n,k-i)) / A000142(k).
EXAMPLE
T(3,1) = 3: 1a1a1a, 2a1a, 3a.
T(3,2) = 4: 1a1a1b, 1a1b1a, 1a1b1b, 2a1b.
T(3,3) = 1: 1a1b1c.
Triangle T(n,k) begins:
1;
0, 1;
0, 2, 1;
0, 3, 4, 1;
0, 5, 12, 7, 1;
0, 7, 30, 33, 11, 1;
0, 11, 72, 130, 77, 16, 1;
0, 15, 160, 463, 438, 157, 22, 1;
0, 22, 351, 1557, 2216, 1223, 289, 29, 1;
0, 30, 743, 5031, 10422, 8331, 2957, 492, 37, 1;
0, 42, 1561, 15877, 46731, 52078, 26073, 6401, 788, 46, 1;
...
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, b[n, i-1, k] + If[i>n, 0, k*b[n-i, i, k]]]]; T[n_, k_] := Sum[b[n, n, k-i]*(-1)^i/(i!*(k-i)!), {i, 0, k}]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 21 2016, after Alois P. Heinz *)
CROSSREFS
Columns k=0-10 give: A000007, A000041 (for n>0), A258457, A258458, A258459, A258460, A258461, A258462, A258463, A258464, A258465.
Row sums give A258466.
T(2n,n) give A258467.
Sequence in context: A300454 A155112 A368099 * A257566 A345117 A188286
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Mar 15 2015
STATUS
approved