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A144645 Triangle in A144643 read upwards by columns.

%I A144645 #21 Jun 09 2025 04:40:33
%S A144645 1,1,0,1,1,0,1,3,1,0,1,6,7,1,0,1,10,25,15,0,0,1,15,65,90,25,0,0,1,21,
%T A144645 140,350,280,35,0,0,1,28,266,1050,1645,770,35,0,0,1,36,462,2646,6825,
%U A144645 6930,1855,0,0,0,1,45,750,5880,22575,39795,26425,3675,0,0,0
%N A144645 Triangle in A144643 read upwards by columns.
%H A144645 G. C. Greubel, <a href="/A144645/b144645.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A144645 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 A144645 David Applegate and N. J. A. Sloane, <a href="http://arxiv.org/abs/0907.0513">The Gift Exchange Problem</a>, arXiv:0907.0513 [math.CO], 2009.
%F A144645 From _G. C. Greubel_, Oct 11 2023: (Start)
%F A144645 T(n, k) = A144643(n-k, n).
%F A144645 T(n, k) = A144644(n, n-k).
%F A144645 T(n, k) = t(n-k, n), where t(n, k) = Sum_{j=0..3} binomial(k-1, j) * t(n-1, k-j-1), with t(n,n) = 1, t(n,k) = 0 if n < 1 or n > k.
%F A144645 Sum_{k=0..n} T(n, k) = A001681(n). (End)
%e A144645 Triangle begins:
%e A144645   1;
%e A144645   1,  0;
%e A144645   1,  1,   0;
%e A144645   1,  3,   1,    0;
%e A144645   1,  6,   7,    1,     0;
%e A144645   1, 10,  25,   15,     0,     0;
%e A144645   1, 15,  65,   90,    25,     0,     0;
%e A144645   1, 21, 140,  350,   280,    35,     0,    0;
%e A144645   1, 28, 266, 1050,  1645,   770,    35,    0,   0;
%e A144645   1, 36, 462, 2646,  6825,  6930,  1855,    0,   0,   0;
%e A144645   1, 45, 750, 5880, 22575, 39795, 26425, 3675,   0,   0,   0;
%t A144645 Table[BellY[n, n-k, {1,1,1,1}], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Oct 11 2023; based on A144644 *)
%o A144645 (Magma)
%o A144645 function t(n,k)
%o A144645   if k eq n then return 1;
%o A144645   elif k le n-1 or n le 0 then return 0;
%o A144645   else return (&+[Binomial(k-1,j)*t(n-1,k-j-1): j in [0..3]]);
%o A144645   end if;
%o A144645 end function;
%o A144645 A144645:= func< n,k | t(n-k,n) >;
%o A144645 [A144645(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Oct 11 2023
%o A144645 (SageMath)
%o A144645 @CachedFunction
%o A144645 def t(n,k):
%o A144645     if (k==n): return 1
%o A144645     elif (k<n or n<1): return 0
%o A144645     else: return sum(binomial(k-1,j)*t(n-1,k-j-1) for j in range(4))
%o A144645 def A144645(n,k): return t(n-k,n)
%o A144645 flatten([[A144645(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Oct 11 2023
%Y A144645 Cf. A001681 (row sums), A144643, A144644.
%K A144645 nonn,tabl
%O A144645 0,8
%A A144645 _David Applegate_ and _N. J. A. Sloane_, Jan 25 2009