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!)
A293024 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(exp(x) - Sum_{i=0..k} x^i/i!). 7
1, 1, 1, 1, 0, 2, 1, 0, 1, 5, 1, 0, 0, 1, 15, 1, 0, 0, 1, 4, 52, 1, 0, 0, 0, 1, 11, 203, 1, 0, 0, 0, 1, 1, 41, 877, 1, 0, 0, 0, 0, 1, 11, 162, 4140, 1, 0, 0, 0, 0, 1, 1, 36, 715, 21147, 1, 0, 0, 0, 0, 0, 1, 1, 92, 3425, 115975, 1, 0, 0, 0, 0, 0, 1, 1, 36, 491, 17722, 678570 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

A(n,k) is the number of set partitions of [n] into blocks of size > k.

LINKS

Seiichi Manyama, Antidiagonals n = 0..139, flattened

E. A. Enneking and J. C. Ahuja, Generalized Bell numbers, Fib. Quart., 14 (1976), 67-73.

FORMULA

E.g.f. of column k: Product_{i>k} exp(x^i/i!).

A(0,k) = 1, A(1,k) = A(2,k) = ... = A(k,k) = 0 and A(n,k) = Sum_{i=k..n-1} binomial(n-1,i)*A(n-1-i,k) for n > k.

EXAMPLE

Square array begins:

    1,  1, 1, 1, 1, ...

    1,  0, 0, 0, 0, ...

    2,  1, 0, 0, 0, ...

    5,  1, 1, 0, 0, ...

   15,  4, 1, 1, 0, ...

   52, 11, 1, 1, 1, ...

MAPLE

A:= proc(n, k) option remember; `if`(n=0, 1, add(

      A(n-j, k)*binomial(n-1, j-1), j=1+k..n))

    end:

seq(seq(A(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Sep 28 2017

MATHEMATICA

A[0, _] = 1;

A[n_, k_] /; 0 <= k <= n := A[n, k] = Sum[A[n-j, k] Binomial[n-1, j-1], {j, k+1, n}];

A[_, _] = 0;

Table[A[n-k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 06 2019 *)

PROG

(Ruby)

def ncr(n, r)

  return 1 if r == 0

  (n - r + 1..n).inject(:*) / (1..r).inject(:*)

end

def A(k, n)

  ary = [1]

  (1..n).each{|i| ary << (k..i - 1).inject(0){|s, j| s + ncr(i - 1, j) * ary[i - 1 - j]}}

  ary

end

def A293024(n)

  a = []

  (0..n).each{|i| a << A(i, n - i)}

  ary = []

  (0..n).each{|i|

    (0..i).each{|j|

      ary << a[i - j][j]

    }

  }

  ary

end

p A293024(20)

CROSSREFS

Columns k=0..5 give A000110, A000296, A006505, A057837, A057814, A293025.

Rows n=0..1 give A000012, A000007.

Main diagonal gives A000007.

Cf. A282988 (as triangle).

Sequence in context: A266972 A266493 A075374 * A292948 A210872 A292973

Adjacent sequences:  A293021 A293022 A293023 * A293025 A293026 A293027

KEYWORD

nonn,tabl

AUTHOR

Seiichi Manyama, Sep 28 2017

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 August 4 09:03 EDT 2020. Contains 336201 sequences. (Running on oeis4.)