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%I #25 Oct 05 2023 08:36:24
%S 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,
%T 1,15,105,420,945,945,0,0,0,0,0,0,1,21,210,1260,4725,10395,10395,0,0,
%U 0,0,0,0,0,1,28,378,3150,17325,62370,135135,135135,0,0,0,0,0,0
%N Triangle b(n,k) for n >= 0, 0 <= k <= 2n, read by rows. See A144299 for definition and properties.
%C 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.
%C Row n has 2n+1 entries.
%C A001498 has a b-file.
%H Reinhard Zumkeller, <a href="/A144331/b144331.txt">Rows n = 0..100 of triangle, flattened</a>
%H Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, <a href="http://arxiv.org/abs/1701.08394">Analysis of the Gift Exchange Problem</a>, arXiv:1701.08394 [math.CO], 2017.
%H David Applegate and N. J. A. Sloane, <a href="http://arxiv.org/abs/0907.0513">The Gift Exchange Problem</a> (arXiv:0907.0513, 2009)
%F E.g.f.: Sum_{n >= 0} Sum_{k = 0..2n} b(n,k) y^n * x^k/k! = exp(x*y*(1 + x/2)).
%F b(n, k) = 2^(n-k)*k!/((2*n-k)!*(k-n)!).
%F Sum_{k=0..2*n} b(n, k) = A001515(n).
%F Sum_{n >= 0} b(n, k) = A000085(k).
%F From _G. C. Greubel_, Oct 04 2023: (Start)
%F 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.
%F Sum_{k=0..2*n} (-1)^k * T(n, k) = A278990(n). (End)
%e Triangle begins:
%e 1
%e 0 1 1
%e 0 0 1 3 3
%e 0 0 0 1 6 15 15
%e 0 0 0 0 1 10 45 105 105
%e 0 0 0 0 0 1 15 105 420 945 945
%e 0 0 0 0 0 0 1 21 210 1260 4725 10395 10395
%e ...
%t 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 *)
%o (Haskell)
%o a144331 n k = a144331_tabf !! n !! k
%o a144331_row n = a144331_tabf !! n
%o a144331_tabf = iterate (\xs ->
%o zipWith (+) ([0] ++ xs ++ [0]) $ zipWith (*) (0:[0..]) ([0,0] ++ xs)) [1]
%o -- _Reinhard Zumkeller_, Nov 24 2014
%o (Magma)
%o A144331:= func< n,k | k le n-1 select 0 else Factorial(k)/(2^(k-n)*Factorial(k-n)*Factorial(2*n-k)) >;
%o [A144331(n,k): k in [0..2*n], n in [0..12]]; // _G. C. Greubel_, Oct 04 2023
%o (SageMath)
%o def A144331(n, k): return 0 if k<n else factorial(k)/(2^(k-n)*factorial(2*n-k)*factorial(k-n))
%o flatten([[A144331(n,k) for k in range(2*n+1)] for n in range(13)]) # _G. C. Greubel_, Oct 04 2023
%Y Cf. A144299, A278990.
%Y Row sums give A001515, column sums A000085.
%Y Other versions of this triangle are given in A001497, A001498, A111924 and A100861.
%Y See A144385 for a generalization.
%K nonn,tabf,nice
%O 0,8
%A _David Applegate_ and _N. J. A. Sloane_, Dec 07 2008
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