login
A330460
Triangle read by rows where T(n,k) is the number of set partitions with k blocks and total sum n.
8
1, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 2, 1, 0, 0, 0, 3, 2, 0, 0, 0, 0, 4, 5, 1, 0, 0, 0, 0, 5, 6, 1, 0, 0, 0, 0, 0, 6, 9, 2, 0, 0, 0, 0, 0, 0, 8, 13, 3, 0, 0, 0, 0, 0, 0, 0, 10, 23, 10, 1, 0, 0, 0, 0, 0, 0, 0, 12, 27, 11, 1, 0, 0, 0, 0, 0, 0, 0, 0, 15, 40, 19, 2, 0, 0, 0, 0, 0, 0, 0, 0
OFFSET
0,8
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows n=0..50)
FORMULA
T(n,k) = Sum_{k <= i <= n} A060016(n,i) * A008277(i,k).
For n > 0, T(n,2) = Sum_{k = 1..n} (2^(k - 1) -1) * A060016(n,k).
EXAMPLE
Triangle begins:
1
0 1
0 1 0
0 2 1 0
0 2 1 0 0
0 3 2 0 0 0
0 4 5 1 0 0 0
0 5 6 1 0 0 0 0
0 6 9 2 0 0 0 0 0
0 8 13 3 0 0 0 0 0 0
0 10 23 10 1 0 0 0 0 0 0
0 12 27 11 1 0 0 0 0 0 0 0
0 15 40 19 2 0 0 0 0 0 0 0 0
Row n = 8 counts the following set partitions:
{{8}} {{1},{7}} {{1},{2},{5}}
{{3,5}} {{2},{6}} {{1},{3},{4}}
{{2,6}} {{3},{5}}
{{1,7}} {{1},{3,4}}
{{1,3,4}} {{1},{2,5}}
{{1,2,5}} {{2},{1,5}}
{{3},{1,4}}
{{4},{1,3}}
{{5},{1,2}}
MAPLE
b:= proc(n, i, k) option remember; `if`(i*(i+1)/2<n, 0,
`if`(n=0, x^k, b(n, i-1, k) +(t-> k*
b(n-i, t, k)+b(n-i, t, k+1))(min(n-i, i-1))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
seq(T(n), n=0..15); # Alois P. Heinz, Dec 29 2019
MATHEMATICA
ppl[n_, k_]:=Switch[k, 0, {n}, 1, IntegerPartitions[n], _, Join@@Table[Union[Sort/@Tuples[ppl[#, k-1]&/@ptn]], {ptn, IntegerPartitions[n]}]];
Table[Length[Select[ppl[n, 2], Length[#]==k&&And[UnsameQ@@#, UnsameQ@@Join@@#]&]], {n, 0, 10}, {k, 0, n}]
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[i(i+1)/2 < n, 0, If[n == 0, x^k, b[n, i-1, k] + Function[t, k*b[n-i, t, k] + b[n-i, t, k + 1]][Min[n-i, i-1]]]];
T[n_] := PadRight[CoefficientList[b[n, n, 0], x], n + 1];
T /@ Range[0, 15] // Flatten (* Jean-François Alcover, May 16 2021, after Alois P. Heinz *)
PROG
(PARI)
A(n)={my(v=Vec(prod(k=1, n, 1 + x^k*y + O(x*x^n)))); vector(#v, n, my(p=v[n]); vector(n, k, sum(i=k, n, polcoef(p, i-1)*stirling(i-1, k-1, 2))))}
{my(T=A(12)); for(n=1, #T, print(T[n]))} \\ Andrew Howroyd, Dec 29 2019
CROSSREFS
Row sums are A294617.
Column k = 1 is A000009 (n > 0).
Sequence in context: A106347 A124300 A154326 * A027186 A131962 A302236
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Dec 18 2019
STATUS
approved