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A124324 Triangle read by rows: T(n,k) is the number of partitions of an n-set having k blocks of size > 1 (0<=k<=floor(n/2)). 13
1, 1, 1, 1, 1, 4, 1, 11, 3, 1, 26, 25, 1, 57, 130, 15, 1, 120, 546, 210, 1, 247, 2037, 1750, 105, 1, 502, 7071, 11368, 2205, 1, 1013, 23436, 63805, 26775, 945, 1, 2036, 75328, 325930, 247555, 27720, 1, 4083, 237127, 1561516, 1939630, 460845, 10395, 1, 8178 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Row sums are the Bell numbers (A000110). T(n,1) = A000295(n) (the Eulerian numbers). Sum(k*T(n,k), k=0..floor(n/2)) = A124325(n).

LINKS

Alois P. Heinz, Rows n = 0..200, flattened

FORMULA

E.g.f.: G(t,z) = exp[t*exp(z)-t+(1-t)z].

T(2n,n) = A001147(n). - Alois P. Heinz, Apr 06 2018

EXAMPLE

T(4,2) = 3 because we have 12|34, 13|24 and 14|23 (if we take {1,2,3,4} as our 4-set).

Triangle starts:

1;

1;

1,    1;

1,    4;

1,   11,     3;

1,   26,    25;

1,   57,   130,    15;

1,  120,   546,   210;

1,  247,  2037,  1750,   105;

1,  502,  7071, 11368,  2205;

1, 1013, 23436, 63805, 26775, 945;

MAPLE

G:=exp(t*exp(z)-t+(1-t)*z): Gser:=simplify(series(G, z=0, 36)): for n from 0 to 33 do P[n]:=sort(n!*coeff(Gser, z, n)) od: for n from 0 to 13 do seq(coeff(P[n], t, k), k=0..floor(n/2)) od; # yields sequence in triangular form

# second Maple program:

b:= proc(n) option remember; expand(`if`(n=0, 1, add(

      `if`(i>1, x, 1)*binomial(n-1, i-1)*b(n-i), i=1..n)))

    end:

T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):

seq(T(n), n=0..15);  # Alois P. Heinz, Mar 08 2015, Jul 15 2017

MATHEMATICA

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] :=  b[n, i] = Expand[If[n == 0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j!*b[n-i*j, i-1]*If[i>1, x^j, 1], {j, 0, n/i}]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-Fran├žois Alcover, May 22 2015, after Alois P. Heinz *)

CROSSREFS

Columns k=0-10 give: A000012, A000295, A112495, A112496, A112497, A290034, A290035, A290036, A290037, A290038, A290039.

Cf. A000110, A001147, A124323, A124325.

Sequence in context: A092288 A111964 A242351 * A178519 A094503 A113897

Adjacent sequences:  A124321 A124322 A124323 * A124325 A124326 A124327

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Oct 28 2006

STATUS

approved

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Last modified July 19 22:14 EDT 2019. Contains 325168 sequences. (Running on oeis4.)