cp's OEIS Frontend

A120057 Table T(n,k) = sum over all set partitions of n of number at index k.

%I A120057 #15 Aug 02 2021 21:30:11
%S A120057 1,2,3,5,8,9,15,25,29,31,52,89,106,115,120,203,354,431,474,499,514,
%T A120057 877,1551,1923,2141,2273,2355,2407,4140,7403,9318,10489,11224,11695,
%U A120057 12002,12205,21147,38154,48635,55286,59595,62434,64331,65614,66491,115975,210803,271617,311469,338019,355951,368205,376665,382559,386699
%N A120057 Table T(n,k) = sum over all set partitions of n of number at index k.
%H A120057 Alois P. Heinz, <a href="/A120057/b120057.txt">Rows n = 1..141, flattened</a>
%F A120057 T(n,k) = Sum_{i=1..k} A120058(n,i)*B(n-i+1), where B(n) are the Bell numbers, (A000110).
%e A120057 The set partitions of 3 are {1,1,1}, {1,1,2}, {1,2,1}, {1,2,2} and {1,2,3}. Summing these componentwise gives the third row: 5,8,9.
%e A120057 Table starts:
%e A120057    1;
%e A120057    2,  3;
%e A120057    5,  8,   9;
%e A120057   15, 25,  29,  31;
%e A120057   52, 89, 106, 115, 120;
%e A120057   ...
%p A120057 b:= proc(n, m) option remember; `if`(n=0, [1, 0],
%p A120057       add((p-> [p[1], expand(p[2]*x+p[1]*j)])(
%p A120057         b(n-1, max(m, j))), j=1..m+1))
%p A120057     end:
%p A120057 T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(b(n, 0)[2]):
%p A120057 seq(T(n), n=1..10);  # _Alois P. Heinz_, Mar 24 2016
%t A120057 b[n_, m_] := b[n, m] = If[n == 0, {1, 0}, Sum[Function[p, {p[[1]], p[[2]]*x + p[[1]]*j}][b[n-1, Max[m, j]]], {j, 1, m+1}]];
%t A120057 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n-1}]][b[n, 0][[2]]];
%t A120057 Table[T[n], {n, 1, 10}] // Flatten (* _Jean-François Alcover_, Apr 08 2016, after _Alois P. Heinz_ *)
%Y A120057 Cf. A120058, A120095. First column is A000110.
%Y A120057 Main diagonal is A087648(n-1).
%Y A120057 Row sums give A346772.
%K A120057 nonn,tabl
%O A120057 1,2
%A A120057 _Franklin T. Adams-Watters_, Jun 06 2006, Jun 07 2006