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

0,8

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.

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)

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)

Example

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
  ...

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 *)

Prog

(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
(SageMath)
def A144331(n, k): return 0 if k<n else factorial(k)/(2^(k-n)*factorial(2*n-k)*factorial(k-n))
flatten([[A144331(n, k) for k in range(2*n+1)] for n in range(13)]) # G. C. Greubel, Oct 04 2023

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.

Sequence in context: A110492 A225413 A180995 * A216805 A167259 A303650

Adjacent sequences: A144328 A144329 A144330 * A144332 A144333 A144334

Keyword

nonn,tabf,nice

Author

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

Status

approved