A144331 Triangle b(n,k) for n >= 0, 0 <= k <= 2n, read by rows. See A144299 for definition and properties.

1, 0, 1, 1, 0, 0, 1, 3, 3, 0, 0, 0, 1, 6, 15, 15, 0, 0, 0, 0, 1, 10, 45, 105, 105, 0, 0, 0, 0, 0, 1, 15, 105, 420, 945, 945, 0, 0, 0, 0, 0, 0, 1, 21, 210, 1260, 4725, 10395, 10395, 0, 0, 0, 0, 0, 0, 0, 1, 28, 378, 3150, 17325, 62370, 135135, 135135, 0, 0, 0, 0, 0, 0

Offset: 0

Author

David Applegate and N. J. A. Sloane, Dec 07 2008

Keywords

• nonn
• tabf
• nice

Comments

Although this entry is the last of the versions of the underlying triangle to be added to the OEIS, for some applications it is the most important.

Row n has 2n+1 entries.

A001498 has a b-file.

Examples

			Triangle begins:
  1
  0 1 1
  0 0 1 3 3
  0 0 0 1 6 15 15
  0 0 0 0 1 10 45 105 105
  0 0 0 0 0  1 15 105 420  945  945
  0 0 0 0 0  0  1  21 210 1260 4725 10395 10395
  ...
		

Links

• Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
• Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
• David Applegate and N. J. A. Sloane, The Gift Exchange Problem (arXiv:0907.0513, 2009)

Crossrefs

Cf. A144299, A278990.

Row sums give A001515, column sums A000085.

Other versions of this triangle are given in A001497, A001498, A111924 and A100861.

See A144385 for a generalization.

Programs

Haskell

a144331 n k = a144331_tabf !! n !! k
a144331_row n = a144331_tabf !! n
a144331_tabf = iterate (\xs ->
  zipWith (+) ([0] ++ xs ++ [0]) $ zipWith (*) (0:[0..]) ([0,0] ++ xs)) [1]
-- Reinhard Zumkeller, Nov 24 2014

Magma

A144331:= func< n,k | k le n-1 select 0 else Factorial(k)/(2^(k-n)*Factorial(k-n)*Factorial(2*n-k)) >;
[A144331(n,k): k in [0..2*n], n in [0..12]]; // G. C. Greubel, Oct 04 2023

Mathematica

Flatten[Table[PadLeft[Table[(n+k)!/(2^k*k!*(n-k)!), {k,0,n}], 2*n+1, 0], {n,0,12}]] (* Jean-François Alcover, Oct 14 2011 *)

SageMath

def A144331(n, k): return 0 if kA144331(n,k) for k in range(2*n+1)] for n in range(13)]) # G. C. Greubel, Oct 04 2023

Formula

E.g.f.: Sum_{n >= 0} Sum_{k = 0..2n} b(n,k) y^n * x^k/k! = exp(x*y*(1 + x/2)).

b(n, k) = 2^(n-k)*k!/((2*n-k)!*(k-n)!).

Sum_{k=0..2*n} b(n, k) = A001515(n).

Sum_{n >= 0} b(n, k) = A000085(k).

From G. C. Greubel, Oct 04 2023: (Start)

T(n, k) = 0 for 0 <= k <= n-1, otherwise T(n, k) = k!/(2^(k-n)*(k-n)!*(2*n-k)!) for n <= k <= 2*n.

Sum_{k=0..2*n} (-1)^k * T(n, k) = A278990(n). (End)