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 A308326 #47 Aug 11 2019 00:09:52
%S A308326 1,1,-1,1,-4,3,1,-13,33,-21,1,-40,270,-546,315,1,-121,2010,-10080,
%T A308326 17955,-9765,1,-364,14433,-165270,707805,-1171800,615195,1,-1093,
%U A308326 102123,-2580081,24421005,-95765355,151953165,-78129765,1,-3280,718140,-39416076,795752370,-6790268520,25331269320,-39221142030,19923090075
%N A308326 The q-analog T(q; n,k) of the triangle A163626 for 0 <= k <= n, for q = 2.
%C A308326 The formulas are given for the general case depending on some fixed integer q. The terms are valid if q = 2.
%C A308326 Special cases: T(0; n,k) = (-1)^k * binomial(n,k) for 0 <= k <= n and T(1; n,k) = A163626(n,k) for 0 <= k <= n.
%F A308326 T(q; n,k) = [k+1]_q * T(q; n-1,k) - [k]_q * T(q; n-1,k-1) for 1 <= k <= n with initial values T(q; n,0) = 1 for n >= 0 and T(q; i,j) = 0 if i < j or j < 0 where [i]_q = (q^i - 1)/(q - 1) for i >= 0.
%F A308326 T(q; n,k) = (1/q^binomial(k+1,2)) * (Sum_{j=0..k} (-1)^j * [k,j]_q * q^binomial(k-j,2) * ([j+1]_q)^n) for 0 <= k <= n and q not equal zero where [m,i]_q are the q-binomials (here A022166 for q = 2) and [i]_q = (q^i - 1)/(q - 1) for i >= 0.
%F A308326 Sum_{k=0..n} T(q; n,k) = A000007(n) for n >= 0.
%F A308326 T(q; n,k)/T(q; k,k) give the q-analogs of the Stirling numbers of the second kind (for q = 2 see A139382, but offset 1).
%F A308326 T(q; n,n) = (-1)^n * Product_{j=1..n} [j]_q for n>=0 with empty product 1 (case n = 0) where [i]_q = (q^i - 1)/(q - 1) for i >= 0.
%F A308326 T(q; n,1) = -[n,1]_(q+1) for n >= 1 where [m,i]_q are the q-binomials (here A022166 for q = 2 and A022167 for q = 3).
%F A308326 G.f. of column k: col(q; t,k) = Sum_{n>=k} T(q; n,k)*t^n = ((-t)^k/(1-t)) * Product_{j=1..k} ([i]_q/(1-[i+1]_q*t)) for k>=0 with empty product 1 (case k=0) and [i]_q = i if q = 1 otherwise (q^i-1)/(q-1) for i>=0.
%e A308326 If q = 2 the triangle T(2; n,k) starts:
%e A308326 n\k: 0 1 2 3 4 5 6 7
%e A308326 ====================================================================
%e A308326 0: 1
%e A308326 1: 1 -1
%e A308326 2: 1 -4 3
%e A308326 3: 1 -13 33 -21
%e A308326 4: 1 -40 270 -546 315
%e A308326 5: 1 -121 2010 -10080 17955 -9765
%e A308326 6: 1 -364 14433 -165270 707805 -1171800 615195
%e A308326 7: 1 -1093 102123 -2580081 24421005 -95765355 151953165 -78129765
%e A308326 etc.
%o A308326 (PARI) q = 2; {T(n,k) = if(k<0 || k>n, 0, if(k==0, 1, if(q==1, (k+1) * T(n-1,k) - k * T(n-1,k-1), ((q^(k+1) - 1)/(q - 1)) * T(n-1,k) - ((q^k - 1)/(q - 1)) * T(n-1,k-1))))};
%o A308326 for(n=0, 9, for(k=0, n, print1(T(n,k), ", "))) \\ _Werner Schulte_, May 26 2019
%Y A308326 Cf. A000007, A007318, A022166, A022167, A139382, A163626.
%K A308326 sign,tabl
%O A308326 0,5
%A A308326 _Werner Schulte_, May 23 2019