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A133607 Triangle read by rows: T(n, k) = qStirling2(n, k, q) for q = -1, with 0 <= k <= n.

%I A133607 #21 Mar 10 2020 09:01:21
%S A133607 1,0,1,0,1,-1,0,1,-1,-1,0,1,-1,-2,1,0,1,-1,-3,2,1,0,1,-1,-4,3,3,-1,0,
%T A133607 1,-1,-5,4,6,-3,-1,0,1,-1,-6,5,10,-6,-4,1,0,1,-1,-7,6,15,-10,-10,4,1,
%U A133607 0,1,-1,-8,7,21,-15,-20,10,5,-1,0,1,-1,-9,8,28,-21,-35,20,15,-5,-1
%N A133607 Triangle read by rows: T(n, k) = qStirling2(n, k, q) for q = -1,  with 0 <= k <= n.
%C A133607 Previous name: Triangle T(n,k), 0<=k<=n, read by rows given by [0, 1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.
%F A133607 Sum_{k, 0<=k<=n}T(n,k)*x^(n-k)= A057077(n), A010892(n), A000012(n), A001519(n), A001835(n), A004253(n), A001653(n), A049685(n-1), A070997(n-1), A070998(n-1), A072256(n), A078922(n), A077417(n-1), A085260(n), A001570(n-1) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 respectively .
%F A133607 Sum_{k, 0<=k<=n}T(n,k)*x^k = A000007(n), A010892(n), A133631(n), A133665(n), A133666(n), A133667(n), A133668(n), A133669(n), A133671(n), A133672(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively .
%F A133607 G.f.: (1-x+y*x)/(1-x+y^2*x^2). - _Philippe Deléham_, Mar 14 2012
%F A133607 T(n,k) = T(n-1,k) - T(n-2,k-2), T(0,0) = T(1,1) = T(2,1) = 1, T(1,0) = T(2,0) = 0, T(2,2) = -1 and T(n,k) = 0 if k<0 or if k>n. - _Philippe Deléham_, Mar 14 2012
%e A133607 Triangle begins:
%e A133607   1;
%e A133607   0, 1;
%e A133607   0, 1, -1;
%e A133607   0, 1, -1, -1;
%e A133607   0, 1, -1, -2, 1;
%e A133607   0, 1, -1, -3, 2, 1;
%e A133607   0, 1, -1, -4, 3, 3, -1;
%e A133607   0, 1, -1, -5, 4, 6, -3, -1;
%e A133607   0, 1, -1, -6, 5, 10, -6, -4, 1;
%e A133607   0, 1, -1, -7, 6, 15, -10, -10, 4, 1;
%e A133607   0, 1, -1, -8, 7, 21, -15, -20, 10, 5, -1;
%e A133607   0, 1, -1, -9, 8, 28, -21, -35, 20, 15, -5, -1;
%e A133607   0, 1, -1, -10, 9, 36, -28, -56, 35, 35, -15, -6, 1;
%e A133607   ...
%e A133607 Triangle A103631 begins:
%e A133607   1;
%e A133607   0, 1;
%e A133607   0, 1, 1;
%e A133607   0, 1, 1, 1;
%e A133607   0, 1, 1, 2, 1;
%e A133607   0, 1, 1, 3, 2, 1;
%e A133607   0, 1, 1, 4, 3, 3, 1;
%e A133607   0, 1, 1, 5, 4, 6, 3, 1;
%e A133607   0, 1, 1, 6, 5, 10, 6, 4, 1;
%e A133607   0, 1, 1, 7, 6, 15, 10, 10, 4, 1;
%e A133607   0, 1, 1, 8, 7, 21, 15, 20, 10, 5, 1;
%e A133607   0, 1, 1, 9, 8, 28, 21, 35, 20, 15, 5, 1;
%e A133607   0, 1, 1, 10, 9, 36, 28, 56, 35, 35, 15, 6, 1;
%e A133607   ...
%e A133607 Triangle A108299 begins:
%e A133607   1;
%e A133607   1, -1;
%e A133607   1, -1, -1;
%e A133607   1, -1, -2, 1;
%e A133607   1, -1, -3, 2, 1;
%e A133607   1, -1, -4, 3, 3, -1;
%e A133607   1, -1, -5, 4, 6, -3, -1;
%e A133607   1, -1, -6, 5, 10, -6, -4, 1;
%e A133607   1, -1, -7, 6, 15, -10, -10, 4, 1;
%e A133607   1, -1, -8, 7, 21, -15, -20, 10, 5, -1;
%e A133607   1, -1, -9, 8, 28, -21, -35, 20, 15, -5, -1;
%e A133607   1, -1, -10, 9, 36, -28, -56, 35, 35, -15, -6, 1;
%e A133607   ...
%t A133607 m = 13
%t A133607 (* DELTA is defined in A084938 *)
%t A133607 DELTA[Join[{0, 1}, Table[0, {m}]], Join[{1, -2, 1}, Table[0, {m}]], m] // Flatten (* _Jean-François Alcover_, Feb 19 2020 *)
%t A133607 qStirling2[n_, k_, q_] /; 1 <= k <= n := q^(k-1) qStirling2[n-1, k-1, q] + Sum[q^j, {j, 0, k-1}] qStirling2[n-1, k, q];
%t A133607 qStirling2[n_, 0, _] := KroneckerDelta[n, 0];
%t A133607 qStirling2[0, k_, _] := KroneckerDelta[0, k];
%t A133607 qStirling2[_, _, _] = 0;
%t A133607 Table[qStirling2[n, k, -1], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Mar 10 2020 *)
%o A133607 (Sage)
%o A133607 from sage.combinat.q_analogues import q_stirling_number2
%o A133607 for n in (0..9):
%o A133607     print([q_stirling_number2(n,k).substitute(q=-1) for k in [0..n]])
%o A133607 # _Peter Luschny_, Mar 09 2020
%Y A133607 Another version is A108299.
%Y A133607 Unsigned version is A103631 (T(n,k) = A103631(n,k)*A057077(k)).
%K A133607 sign,tabl
%O A133607 0,14
%A A133607 _Philippe Deléham_, Dec 27 2007
%E A133607 New name from _Peter Luschny_, Mar 09 2020