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%I #18 May 16 2021 12:24:54
%S 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,
%T 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,
%U 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
%N Triangle read by rows where T(n,k) is the number of set partitions with k blocks and total sum n.
%H Andrew Howroyd, <a href="/A330460/b330460.txt">Table of n, a(n) for n = 0..1325</a> (rows n=0..50)
%F T(n,k) = Sum_{k <= i <= n} A060016(n,i) * A008277(i,k).
%F For n > 0, T(n,2) = Sum_{k = 1..n} (2^(k - 1) -1) * A060016(n,k).
%e Triangle begins:
%e 1
%e 0 1
%e 0 1 0
%e 0 2 1 0
%e 0 2 1 0 0
%e 0 3 2 0 0 0
%e 0 4 5 1 0 0 0
%e 0 5 6 1 0 0 0 0
%e 0 6 9 2 0 0 0 0 0
%e 0 8 13 3 0 0 0 0 0 0
%e 0 10 23 10 1 0 0 0 0 0 0
%e 0 12 27 11 1 0 0 0 0 0 0 0
%e 0 15 40 19 2 0 0 0 0 0 0 0 0
%e Row n = 8 counts the following set partitions:
%e {{8}} {{1},{7}} {{1},{2},{5}}
%e {{3,5}} {{2},{6}} {{1},{3},{4}}
%e {{2,6}} {{3},{5}}
%e {{1,7}} {{1},{3,4}}
%e {{1,3,4}} {{1},{2,5}}
%e {{1,2,5}} {{2},{1,5}}
%e {{3},{1,4}}
%e {{4},{1,3}}
%e {{5},{1,2}}
%p b:= proc(n, i, k) option remember; `if`(i*(i+1)/2<n, 0,
%p `if`(n=0, x^k, b(n, i-1, k) +(t-> k*
%p b(n-i, t, k)+b(n-i, t, k+1))(min(n-i, i-1))))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
%p seq(T(n), n=0..15); # _Alois P. Heinz_, Dec 29 2019
%t ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];
%t Table[Length[Select[ppl[n,2],Length[#]==k&&And[UnsameQ@@#,UnsameQ@@Join@@#]&]],{n,0,10},{k,0,n}]
%t (* Second program: *)
%t 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 T[n_] := PadRight[CoefficientList[b[n, n, 0], x], n + 1];
%t T /@ Range[0, 15] // Flatten (* _Jean-François Alcover_, May 16 2021, after _Alois P. Heinz_ *)
%o (PARI)
%o 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))))}
%o {my(T=A(12)); for(n=1, #T, print(T[n]))} \\ _Andrew Howroyd_, Dec 29 2019
%Y Row sums are A294617.
%Y Column k = 1 is A000009 (n > 0).
%Y Cf. A000110, A008277, A050342, A060016, A072706, A270995, A271619, A279375, A279785, A326701, A330459, A330462, A330463, A330759.
%K nonn,tabl
%O 0,8
%A _Gus Wiseman_, Dec 18 2019
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