login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A144299 Triangle of Bessel numbers read by rows. Row n gives T(n,n), T(n,n-1), T(n,n-2), ..., T(n,0) for n >= 0. 10
1, 1, 0, 1, 1, 0, 1, 3, 0, 0, 1, 6, 3, 0, 0, 1, 10, 15, 0, 0, 0, 1, 15, 45, 15, 0, 0, 0, 1, 21, 105, 105, 0, 0, 0, 0, 1, 28, 210, 420, 105, 0, 0, 0, 0, 1, 36, 378, 1260, 945, 0, 0, 0, 0, 0, 1, 45, 630, 3150, 4725, 945, 0, 0, 0, 0, 0, 1, 55, 990, 6930, 17325, 10395, 0, 0, 0, 0, 0, 0, 1, 66, 1485, 13860, 51975, 62370 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

T(n,k) = number of partitions of an n-set into k nonempty subsets, each of size at most 2.

The Grosswald and Choi-Smith references give many further properties and formulas.

Considered as an infinite lower triangular matrix T, Lim_{n->inf.} T^n = A118930: (1, 1, 2, 4, 13, 41, 166, 652, ...) as a vector. [Gary W. Adamson, Dec 08 2008]

A001498 has a b-file.

REFERENCES

E. Grosswald, Bessel Polynomials, Lecture Notes Math., Vol. 698, 1978.

LINKS

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.

J. Y. Choi and J. D. H. Smith, On the unimodality and combinatorics of Bessel numbers, Discrete Math., 264 (2003), 45-53.

T. Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras

Toufik Mansour, Matthias Schork and Mark Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.

T. Mansour and M. Shattuck, Partial matchings and pattern avoidance, Appl. Anal. Discrete Math. 7 (2013) 25-50.

FORMULA

T(n, k) = T(n-1, k-1) + (n-1)*T(n-2, k-1).

E.g.f.: Sum_{k >= 0} Sum_{n = 0..2k} T(n,k) y^k x^n/n! = exp(y(x+x^2/2)). (The coefficient of y^k is the e.g.f. for the k-th row of the rotated triangle shown below.)

EXAMPLE

Triangle begins:

n:

0: 1

1: 1 0

2: 1 1 0

3: 1 3 0 0

4: 1 6 3 0 0

5: 1 10 15 0 0 0

6: 1 15 45 15 0 0 0

7: 1 21 105 105 0 0 0 0

8: 1 28 210 420 105 0 0 0 0

9: 1 36 378 1260 945 0 0 0 0 0

...

The row sums give A000085.

For some purposes it is convenient to rotate the triangle by 45 degrees:

1.0.0.0.0..0..0...0...0....0....0.....0....

..1.1.0.0..0..0...0...0....0....0.....0....

....1.3.3..0..0...0...0....0....0.....0....

......1.6.15.15...0...0....0....0.....0....

........1.10.45.105.105....0....0.....0....

...........1.15.105.420..945..945.....0....

..............1..21.210.1260.4725.10395....

..................1..28..378.3150.17325....

......................1...36..630..6930....

...........................1...45...990....

...

The latter triangle is important enough that it has its own entry, A144331. Here the column sums give A000085 and the rows sums give A001515.

If the entries in the rotated triangle are denoted by b1(n,k), n >= 0, k <= 2n, the we have the recurrence b1(n, k) = b1(n - 1, k - 1) + (k - 1)*b1(n - 1, k - 2).

Then b1(n,k) is the number of partitions of [1, 2, ...,k] into exactly n blocks, each of size 1 or 2.

MAPLE

Maple code producing the rotated version:

b1 := proc(n, k)

option remember;

if n = k then 1;

elif k < n then 0;

elif n < 1 then 0;

else b1(n - 1, k - 1) + (k - 1)*b1(n - 1, k - 2);

end if;

end proc;

for n from 0 to 12 do lprint([seq(b1(n, k), k=0..2*n)]); od:

MATHEMATICA

T[n_, 0] = 0; T[1, 1] = 1; T[2, 1] = 1; T[n_, k_] := T[n - 1, k - 1] + (n - 1)T[n - 2, k - 1]; Table[T[n, k], {n, 12}, {k, n, 1, -1}] // Flatten (* Robert G. Wilson v *)

PROG

(Haskell)

a144299 n k = a144299_tabl !! n !! k

a144299_row n = a144299_tabl !! n

a144299_tabl = [1] : [1, 0] : f 1 [1] [1, 0] where

   f i us vs = ws : f (i + 1) vs ws where

               ws = (zipWith (+) (0 : map (i *) us) vs) ++ [0]

-- Reinhard Zumkeller, Jan 01 2014

CROSSREFS

Other versions of this same triangle are given in A111924 (which omits the first row), A001498 (which left-adjusts the rows in the bottom view), A001497 and A100861. Row sums give A000085.

Cf. A144385, A144643.

Cf. A118930. [Gary W. Adamson, Dec 08 2008]

Sequence in context: A211649 A202023 A080159 * A060514 A176788 A238129

Adjacent sequences:  A144296 A144297 A144298 * A144300 A144301 A144302

KEYWORD

nonn,tabl,easy

AUTHOR

David Applegate and N. J. A. Sloane, Dec 06 2008

EXTENSIONS

Offset fixed by Reinhard Zumkeller, Jan 01 2014

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
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 27 17:25 EDT 2020. Contains 338035 sequences. (Running on oeis4.)