This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A144502 #42 Oct 08 2023 09:04:36
%S A144502 1,1,1,2,2,1,7,7,5,1,37,37,30,16,1,266,266,229,155,65,1,2431,2431,
%T A144502 2165,1633,946,326,1,27007,27007,24576,19714,13219,6687,1957,1,353522,
%U A144502 353522,326515,272501,198773,119917,53822,13700,1,5329837,5329837,4976315,4269271,3289726,2199722,1205857,486355,109601,1
%N A144502 Square array read by antidiagonals upwards: T(n,k) is the number of scenarios for the gift exchange problem in which each gift can be stolen at most once, when there are n gifts in the pool and k gifts (not yet frozen) in peoples' hands.
%H A144502 G. C. Greubel, <a href="/A144502/b144502.txt">Antidiagonals n = 0..50, flattened</a>
%H A144502 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 A144502 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 A144502 Let A_n(x) be the e.g.f. for row n. Then A_0(x) = exp(x) and for n >= 1, A_n(x) = (d/dx)A_{n-1}(x)/(1-x).
%F A144502 For n >= 1, the rows A_{n}(x) = P_{n}(x)*exp(x)/(1-x)^(2*n), where P_{n}(x) = (1-x)*(d/dx)( P_{n-1}(x) ) + (2*n-x)*P_{n-1}(x) and P_{0}(x) = 1. - _G. C. Greubel_, Oct 08 2023
%e A144502 The array, A(n,k), begins:
%e A144502 1, 1, 1, 1, 1, 1, ...
%e A144502 1, 2, 5, 16, 65, 326, ...
%e A144502 2, 7, 30, 155, 946, 6687, ...
%e A144502 7, 37, 229, 1633, 13219, 119917, ...
%e A144502 37, 266, 2165, 19714, 198773, 2199722, ...
%e A144502 266, 2431, 24576, 272501, 3289726, 42965211, ...
%e A144502 ...
%e A144502 Antidiagonal triangle, T(n,k), begins as:
%e A144502 1;
%e A144502 1, 1;
%e A144502 2, 2, 1;
%e A144502 7, 7, 5, 1;
%e A144502 37, 37, 30, 16, 1;
%e A144502 266, 266, 229, 155, 65, 1;
%e A144502 2431, 2431, 2165, 1633, 946, 326, 1;
%e A144502 27007, 27007, 24576, 19714, 13219, 6687, 1957, 1;
%p A144502 B:=proc(p,r) option remember;
%p A144502 if p=0 then RETURN(1); fi;
%p A144502 if r=0 then RETURN(B(p-1,1)); fi;
%p A144502 B(p-1,r+1)+r*B(p,r-1); end;
%p A144502 seq(seq(B(d-k, k), k=0..d), d=0..9);
%t A144502 t[0, _]= 1; t[n_, 0]:= t[n, 0]= t[n-1, 1];
%t A144502 t[n_, k_]:= t[n, k]= t[n-1, k+1] + k*t[n, k-1];
%t A144502 Table[t[n-k, k], {n,0,12}, {k,0,n}]//Flatten (* _Jean-François Alcover_, Jan 14 2014, after Maple *)
%o A144502 (Magma)
%o A144502 A144301:= func< n | (&+[ Binomial(n+k-1,2*k)*Factorial(2*k)/( Factorial(k)*2^k): k in [0..n]]) >;
%o A144502 function A(n,k)
%o A144502 if n eq 0 then return 1;
%o A144502 elif k eq 0 then return A144301(n);
%o A144502 elif k eq 1 then return A144301(n+1);
%o A144502 else return A(n-1,k+1) + k*A(n,k-1);
%o A144502 end if;
%o A144502 end function;
%o A144502 A144502:= func< n,k | A(n-k, k) >;
%o A144502 [A144502(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Sep 29 2023
%o A144502 (SageMath)
%o A144502 def A144301(n): return 1 if n<2 else (2*n-3)*A144301(n-1)+A144301(n-2)
%o A144502 @CachedFunction
%o A144502 def A(n,k):
%o A144502 if n==0: return 1
%o A144502 elif k==0: return A144301(n)
%o A144502 elif k==1: return A144301(n+1)
%o A144502 else: return A(n-1,k+1) + k*A(n,k-1)
%o A144502 def A144502(n,k): return A(n-k,k)
%o A144502 flatten([[A144502(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Sep 29 2023
%Y A144502 Rows include A000522, A144495, A144496, A144497.
%Y A144502 Columns include A144301, A001515, A144498, A144499, A144500.
%Y A144502 Main diagonal is A144501.
%Y A144502 Antidiagonal sums give A144503.
%K A144502 nonn,tabl
%O A144502 0,4
%A A144502 _David Applegate_ and _N. J. A. Sloane_, Dec 13 2008
%E A144502 6 more terms from _Michel Marcus_, Feb 01 2023