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A285595 Sum T(n,k) of the k-th entries in all blocks of all set partitions of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows. 8
1, 4, 2, 17, 10, 3, 76, 52, 18, 4, 362, 274, 111, 28, 5, 1842, 1500, 675, 200, 40, 6, 9991, 8614, 4185, 1380, 325, 54, 7, 57568, 51992, 26832, 9568, 2510, 492, 70, 8, 351125, 329650, 178755, 67820, 19255, 4206, 707, 88, 9, 2259302, 2192434, 1239351, 494828, 149605, 35382, 6629, 976, 108, 10 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

T(n,k) is also k times the number of blocks of size >k in all set partitions of [n+1]. T(3,2) = 10 = 2 * 5 because there are 5 blocks of size >2 in all set partitions of [4], namely in 1234, 123|4, 124|3, 134|2, 1|234.

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

Wikipedia, Partition of a set

FORMULA

T(n,k) = k * Sum_{j=k+1..n+1} binomial(n+1,j)*A000110(n+1-j).

T(n,k) = k * Sum_{j=k+1..n+1} A175757(n+1,j).

Sum_{k=1..n} T(n,k)/k = A278677(n+1).

EXAMPLE

T(3,2) = 10 because the sum of the second entries in all blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 2+2+3+3+0  = 10.

Triangle T(n,k) begins:

:     1;

:     4,     2;

:    17,    10,     3;

:    76,    52,    18,    4;

:   362,   274,   111,   28,    5;

:  1842,  1500,   675,  200,   40,   6;

:  9991,  8614,  4185, 1380,  325,  54,  7;

: 57568, 51992, 26832, 9568, 2510, 492, 70, 8;

MAPLE

T:= proc(h) option remember; local b; b:=

      proc(n, l) option remember; `if`(n=0, [1, 0],

        (p-> p+[0, (h-n+1)*p[1]*x^1])(b(n-1, [l[], 1]))+

         add((p-> p+[0, (h-n+1)*p[1]*x^(l[j]+1)])(b(n-1,

         sort(subsop(j=l[j]+1, l), `>`))), j=1..nops(l)))

      end: (p-> seq(coeff(p, x, i), i=1..n))(b(h, [])[2])

    end:

seq(T(n), n=1..12);

# second Maple program:

b:= proc(n) option remember; `if`(n=0, [1, 0],

      add((p-> p+[0, p[1]*add(x^k, k=1..j-1)])(

         b(n-j)*binomial(n-1, j-1)), j=1..n))

    end:

T:= n-> (p-> seq(coeff(p, x, i)*i, i=1..n))(b(n+1)[2]):

seq(T(n), n=1..12);

MATHEMATICA

b[n_] := b[n] = If[n == 0, {1, 0}, Sum[# + {0, #[[1]]*Sum[x^k, {k, 1, j-1} ]}&[b[n - j]*Binomial[n - 1, j - 1]], {j, 1, n}]];

T[n_] := Table[Coefficient[#, x, i]*i, {i, 1, n}] &[b[n + 1][[2]]];

Table[T[n], {n, 1, 12}] // Flatten (* Jean-Fran├žois Alcover, May 23 2018, translated from 2nd Maple program *)

CROSSREFS

Column k=1 gives A124325(n+1).

Row sums give A000110(n) * A000217(n) = A105488(n+3).

Main diagonal and first lower diagonal give: A000027, A028552.

Cf. A007318, A175757, A278677, A283424, A285362, A285793, A286897.

Sequence in context: A122749 A189741 A303142 * A255566 A302461 A303243

Adjacent sequences:  A285592 A285593 A285594 * A285596 A285597 A285598

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Apr 22 2017

STATUS

approved

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Last modified September 17 13:00 EDT 2019. Contains 327131 sequences. (Running on oeis4.)