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A330460 Triangle read by rows where T(n,k) is the number of set partitions with k blocks and total sum n.

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