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 A334014 #38 May 11 2020 02:19:29
%S A334014 1,1,0,1,1,0,1,2,3,2,1,3,8,18,30,1,4,15,52,163,444,1,5,24,110,478,
%T A334014 1950,7360,1,6,35,198,1083,5706,28821,138690,1,7,48,322,2110,13482,
%U A334014 83824,505876,2954364,1,8,63,488,3715,27768,203569,1461944,10270569,70469000,1,9,80,702,6078,51894,436656,3618540,29510268,236644092,1864204416,1,10,99,970,9403,90150,854485,8003950,74058105,676549450,6098971555,54224221050
%N A334014 Array read by antidiagonals: T(n,k) is the number of functions f: X->Y, where X is a subset of Y, |X| = n, |Y| = n+k, such that for every x in X, f(f(x)) != x.
%C A334014 Comes up in the study of the Zen Stare game (see description at A134362).
%C A334014 T(k,n-k)*binomial(n,k)*(n-k-1)!! is the number of different possible Zen Stare rounds with n starting players and k winners.
%H A334014 Andrew Howroyd, <a href="/A334014/b334014.txt">Table of n, a(n) for n = 0..1325</a>
%F A334014 T(n,k) = Sum_{i=0..n} k^(n-i)*binomial(n,i)*T(i,n-i); This means that with a constant n, T(n,k) is a polynomial of k.
%F A334014 T(n,0) = A134362(n).
%F A334014 T(0,k) = 1.
%F A334014 For odd n, Sum_{k=1..(n+1)/2} T(2*k-1,n-2*k+1)*binomial(n,2*k-1)*(n-2*k)!! = (n-1)^n.
%F A334014 E.g.f. of k-th column: exp((k-1)*W(x) - W(x)^2/2)/(1-W(x)) where W(x) is the e.g.f. of A000169. - _Andrew Howroyd_, Apr 15 2020
%e A334014 Array begins:
%e A334014 =======================================================
%e A334014 n\k | 0 1 2 3 4 5 6
%e A334014 ----+--------------------------------------------------
%e A334014 0 | 1 1 1 1 1 1 1 ...
%e A334014 1 | 0 1 2 3 4 5 6 ...
%e A334014 2 | 0 3 8 15 24 35 48 ...
%e A334014 3 | 2 18 52 110 198 322 488 ...
%e A334014 4 | 30 163 478 1083 2110 3715 6078 ...
%e A334014 5 | 444 1950 5706 13482 27768 51894 90150 ...
%e A334014 6 | 7360 28821 83824 203569 436656 854485 1557376 ...
%e A334014 ...
%e A334014 T(2,2) = 8; This because given X = {A,B}, Y = {A,B,C,D}. The only functions f: X->Y that meet the requirement are:
%e A334014 f(A) = C, f(B) = C
%e A334014 f(A) = D, f(B) = D
%e A334014 f(A) = D, f(B) = C
%e A334014 f(A) = C, f(B) = D
%e A334014 f(A) = B, f(B) = C
%e A334014 f(A) = B, f(B) = D
%e A334014 f(A) = C, f(B) = A
%e A334014 f(A) = D, f(B) = A
%o A334014 (PARI) T(n,k)={my(w=-lambertw(-x + O(x^max(4,1+n)))); n!*polcoef(exp((k-1)*w - w^2/2)/(1-w), n)} \\ _Andrew Howroyd_, Apr 15 2020
%Y A334014 Rows n=0..3 are A000012, A001477, A005563, A058794.
%Y A334014 Columns k=0..4 are A134362, A089466, A089467, A089468, A220690(n+2).
%Y A334014 Cf. A000169, A065440.
%K A334014 nonn,tabl
%O A334014 0,8
%A A334014 _Mason C. Hart_, Apr 14 2020