A285410 -id:A285410 - cp's OEIS frontend

A285362 Sum T(n,k) of the entries in the k-th blocks of all set partitions of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

1, 4, 2, 15, 12, 3, 60, 58, 28, 4, 262, 273, 185, 55, 5, 1243, 1329, 1094, 495, 96, 6, 6358, 6839, 6293, 3757, 1148, 154, 7, 34835, 37423, 36619, 26421, 11122, 2380, 232, 8, 203307, 217606, 219931, 180482, 96454, 28975, 4518, 333, 9, 1257913, 1340597, 1376929, 1230737, 787959, 308127, 67898, 7995, 460, 10

Offset: 1

Alois P. Heinz, Apr 17 2017

			T(3,2) = 12 because the sum of the entries in the second blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 0+3+2+5+2 = 12.
Triangle T(n,k) begins:
      1;
      4,     2;
     15,    12,     3;
     60,    58,    28,     4;
    262,   273,   185,    55,     5;
   1243,  1329,  1094,   495,    96,    6;
   6358,  6839,  6293,  3757,  1148,  154,   7;
  34835, 37423, 36619, 26421, 11122, 2380, 232, 8;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Wikipedia, Partition of a set

Columns k=1-10 give: A285363, A285364, A285365, A285366, A285367, A285368, A285369, A285370, A285371, A285372.
Row sums give A000110(n) * A000217(n) = A105488(n+3).
Main diagonal and first lower diagonal give: A000027, A006000 (for n>0).
T(2n+1,n+1) gives A285410.
Cf. A270236, A285439, A285595.

T:= proc(h) option remember; local b; b:=
      proc(n, m) option remember; `if`(n=0, [1, 0], add((p-> p
        +[0, (h-n+1)*p[1]*x^j])(b(n-1, max(m, j))), j=1..m+1))
      end: (p-> seq(coeff(p, x, i), i=1..n))(b(h, 0)[2])
    end:
seq(T(n), n=1..12);

T[h_] := T[h] = Module[{b}, b[n_, m_] := b[n, m] = If[n == 0, {1, 0}, Sum[# + {0, (h - n + 1)*#[[1]]*x^j}&[b[n - 1, Max[m, j]]], {j, 1, m + 1}]]; Table[Coefficient[#, x, i], {i, 1, n}]&[b[h, 0][[2]]]];
Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

A270529 Sum of the sizes of the (n+1)-th blocks in all set partitions of {1,2,...,2n+1}.

Original entry on oeis.org

1, 5, 47, 675, 13276, 334751, 10354804, 380797185, 16262852622, 792102157717, 43370872479317, 2638621340623857, 176656418678888190, 12910491906798508171, 1022900642521227415940, 87345042902079159197907, 7997120745886569461943400, 781580696472700788364550933

Offset: 0

Alois P. Heinz, Mar 18 2016

nonn

			a(1) = 5 = 0+1+1+2+1 = sum of the sizes of the second blocks in all A000110(3) = 5 set partitions of 3: 123, 12|3, 13|2, 1|23, 1|2|3.

Alois P. Heinz, Table of n, a(n) for n = 0..200
Wikipedia, Partition of a set

Cf. A000110, A270236, A285410.

b:= proc(n, m, k) option remember; `if`(n=0, [1, 0], add((p->p+
      `if`(j=k, [0, p[1]], 0))(b(n-1, max(m, j), k)), j=1..m+1))
    end:
a:= n-> b(2*n+1, 0, n+1)[2]:
seq(a(n), n=0..20);

b[n_, m_, k_] := b[n, m, k] = If[n == 0, {1, 0}, Sum[# + If[j == k, {0, #[[1]]}, 0]&[b[n - 1, Max[m, j], k]], {j, 1, m + 1}]];
a[n_] := b[2*n + 1, 0, n + 1][[2]];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 23 2018, translated from Maple *)

a(n) = A270236(2n+1,n+1).

a(n) ~ 2^(2*n+1/2) * n^(n-1/2) / (sqrt(Pi*(1-c)) * exp(n) * c^(n+1) * (2-c)^n), where c = -A226775 = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - Vaclav Kotesovec, Mar 19 2016

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A285362 Sum T(n,k) of the entries in the k-th blocks of all set partitions of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

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