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 A274105 #55 Feb 16 2025 08:33:36
%S A274105 1,1,1,1,2,1,5,4,1,8,14,4,1,13,46,46,8,1,18,98,184,100,8,1,25,206,674,
%T A274105 836,308,16,1,32,356,1704,3532,2816,632,16,1,41,612,4196,13756,20476,
%U A274105 11896,1912,32,1,50,940,8480,38932,89256,93800,37600,3856,32,1,61,1440,16940,106772,361780,629336,506600,154256,11600,64
%N A274105 Triangle read by rows: T(n,k) = number of configurations of k nonattacking bishops on the black squares of an n X n chessboard (0 <= k <= n - [n>1]).
%C A274105 Rows give the coefficients of the independence polynomial of the n X n black bishop graph. - _Eric W. Weisstein_, Jun 26 2017
%H A274105 Irving Kaplansky and John Riordan, <a href="http://projecteuclid.org/euclid.dmj/1077473616">The problem of the rooks and its applications</a>, Duke Mathematical Journal 13.2 (1946): 259-268. See Section 9.
%H A274105 Irving Kaplansky and John Riordan, <a href="/A274105/a274105.pdf">The problem of the rooks and its applications</a>, in Combinatorics, Duke Mathematical Journal, 13.2 (1946): 259-268. See Section 9. [Annotated scanned copy]
%H A274105 Eder G. Santos, <a href="https://arxiv.org/abs/2411.16492">Counting non-attacking chess pieces placements: Bishops and Anassas</a>. arXiv:2411.16492 [math.CO], 2024. (considered as white board).
%H A274105 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BlackBishopGraph.html">Black Bishop Graph</a>
%H A274105 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IndependencePolynomial.html">Independence Polynomial</a>
%F A274105 From _Eder G. Santos_, Dec 01 2024: (Start)
%F A274105 T(n,k) = Sum_{j=0..k} binomial(ceiling(n/2),j) * Stirling2(n-j,n-k).
%F A274105 T(n,k) = T(n-1,k) + (n-k+A000035(n)) * T(n-1,k-1), T(n,0) = 1, T(0,k) = delta(k,0). (End)
%e A274105 Triangle begins:
%e A274105 1;
%e A274105 1, 1;
%e A274105 1, 2;
%e A274105 1, 5, 4;
%e A274105 1, 8, 14, 4;
%e A274105 1, 13, 46, 46, 8;
%e A274105 1, 18, 98, 184, 100, 8;
%e A274105 1, 25, 206, 674, 836, 308, 16;
%e A274105 1, 32, 356, 1704, 3532, 2816, 632, 16;
%e A274105 1, 41, 612, 4196, 13756, 20476, 11896, 1912, 32;
%e A274105 1, 50, 940, 8480, 38932, 89256, 93800, 37600, 3856, 32;
%e A274105 1, 61, 1440, 16940, 106772, 361780, 629336, 506600, 154256, 11600, 64;
%e A274105 ...
%e A274105 Corresponding independence polynomials:
%e A274105 1, (empty graph)
%e A274105 1+x, (K_1)
%e A274105 1+2*x, (P_2 = K_2)
%e A274105 1+5*x+4*x^2, (butterfly graph)
%e A274105 1+8*x+14*x^2+4*x^3,
%e A274105 ...
%p A274105 with(combinat); with(gfun);
%p A274105 T:=n->add(stirling2(n+1,n+1-k)*x^k, k=0..n);
%p A274105 # bishops on black squares
%p A274105 bish:=proc(n) local m,k,i,j,t1,t2; global T;
%p A274105 if n<2 then return [1$(n+1)] fi;
%p A274105 if (n mod 2) = 0 then m:=n/2;
%p A274105 t1:=add(binomial(m,k)*T(2*m-1-k)*x^k, k=0..m);
%p A274105 else
%p A274105 m:=(n-1)/2;
%p A274105 t1:=add(binomial(m+1,k)*T(2*m-k)*x^k, k=0..m+1);
%p A274105 fi;
%p A274105 seriestolist(series(t1,x,2*n+1));
%p A274105 end;
%p A274105 for n from 0 to 12 do lprint(bish(n)); od:
%p A274105 # second Maple program:
%p A274105 T:= (n,k)-> add(binomial(ceil(n/2),j)*Stirling2(n-j,n-k),j=0..k):
%p A274105 seq(seq(T(n,k), k=0..n-`if`(n>1,1,0)), n=0..11); # _Alois P. Heinz_, Dec 01 2024
%t A274105 CoefficientList[Table[Sum[x^n Binomial[Ceiling[n/2], k] BellB[n - k, 1/x], {k, 0, Ceiling[n/2]}], {n, 10}], x] (* _Eric W. Weisstein_, Jun 26 2017 *)
%o A274105 (SageMath) def stirling2_negativek(n, k):
%o A274105 if k < 0: return 0
%o A274105 else: return stirling_number2(n, k)
%o A274105 print([sum([binomial(ceil(n/2), l)*stirling2_negativek(n-l, n-k) for l in [0..k]]) for n in [0..10] for k in [0..n-1+kronecker_delta(n,1)+kronecker_delta(n,0)]]) # _Eder G. Santos_, Dec 01 2024
%Y A274105 Alternate rows give A088960.
%Y A274105 Row sums are A216332(n+1).
%Y A274105 Cf. A274106 (white squares), A288183, A201862, A002465.
%K A274105 nonn,tabf
%O A274105 0,5
%A A274105 _N. J. A. Sloane_, Jun 14 2016
%E A274105 T(0,0) prepended by _Eder G. Santos_, Dec 01 2024