login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A124323 Triangle read by rows: T(n,k) is the number of partitions of an n-set having k singleton blocks (0<=k<=n). 11
1, 0, 1, 1, 0, 1, 1, 3, 0, 1, 4, 4, 6, 0, 1, 11, 20, 10, 10, 0, 1, 41, 66, 60, 20, 15, 0, 1, 162, 287, 231, 140, 35, 21, 0, 1, 715, 1296, 1148, 616, 280, 56, 28, 0, 1, 3425, 6435, 5832, 3444, 1386, 504, 84, 36, 0, 1, 17722, 34250, 32175, 19440, 8610, 2772, 840, 120, 45, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Row sums are the Bell numbers (A000110). T(n,0)=A000296(n). T(n,k) = binomial(n,k)*T(n-k,0). Sum(k*T(n,k),k=0..n) = A052889(n) = n*B(n-1), where B(q) are the Bell numbers (A000110).

Exponential Riordan array [exp(exp(x)-1-x),x]. - Paul Barry, Apr 23 2009

Sum_{k=0..n} T(n,k)*2^k = A000110(n+1) is the number of binary relations on an n-set that are both symmetric and transitive. - Geoffrey Critzer, Jul 25 2014

REFERENCES

T. Mansour, A. O. Munagi, Set partitions with circular successions, European Journal of Combinatorics, 42 (2014), 207-216.

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

FORMULA

T(n,k) = binomial(n,k)*[(-1)^(n-k)+sum((-1)^(j-1)*B(n-k-j), j=1..n-k)], where B(q) are the Bell numbers (A000110).

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

G.f.: 1/(1-xy-x^2/(1-xy-x-2x^2/(1-xy-2x-3x^2/(1-xy-3x-4x^2/(1-... (continued fraction). - Paul Barry, Apr 23 2009

EXAMPLE

T(4,2)=6 because we have 12|3|4, 13|2|4, 14|2|3, 1|23|4, 1|24|3 and 1|2|34 (if we take {1,2,3,4} as our 4-set).

Triangle starts:

1;

0,   1;

1,   0,  1;

1,   3,  0,  1;

4,   4,  6,  0, 1;

11, 20, 10, 10, 0, 1;

...

From Paul Barry, Apr 23 2009: (Start)

Production matrix is

0, 1,

1, 0, 1,

1, 2, 0, 1,

1, 3, 3, 0, 1,

1, 4, 6, 4, 0, 1,

1, 5, 10, 10, 5, 0, 1,

1, 6, 15, 20, 15, 6, 0, 1,

1, 7, 21, 35, 35, 21, 7, 0, 1,

1, 8, 28, 56, 70, 56, 28, 8, 0, 1 (End)

MAPLE

G:=exp(exp(z)-1+(t-1)*z): Gser:=simplify(series(G, z=0, 14)): for n from 0 to 11 do P[n]:=sort(n!*coeff(Gser, z, n)) od: for n from 0 to 11 do seq(coeff(P[n], t, k), k=0..n) od; # yields sequence in triangular form

# Program from R. J. Mathar, Jan 22 2015:

A124323 := proc(n, k)

    binomial(n, k)*A000296(n-k) ;

end proc:

MATHEMATICA

Flatten[CoefficientList[Range[0, 10]! CoefficientList[Series[Exp[x y] Exp[Exp[x] - x - 1], {x, 0, 10}], x], y]] (* Geoffrey Critzer, Nov 24 2011 *)

CROSSREFS

Cf. A000110, A052889, A124324.

A250104 is an essentially identical triangle, differing only in row 1.

For columns see A000296, A250105, A250106, A250107.

Sequence in context: A137176 A143949 * A250104 A220421 A106683 A139601

Adjacent sequences:  A124320 A124321 A124322 * A124324 A124325 A124326

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Oct 28 2006

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 20 22:38 EDT 2018. Contains 316404 sequences. (Running on oeis4.)