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 A306344 #10 Feb 11 2019 13:28:55
%S A306344 1,1,1,1,2,1,1,5,5,1,1,14,17,14,1,1,41,53,53,41,1,1,122,161,170,161,
%T A306344 122,1,1,365,485,521,521,485,365,1,1,1094,1457,1574,1601,1574,1457,
%U A306344 1094,1,1,3281,4373,4733,4841,4841,4733,4373,3281,1
%N A306344 The q-analogs T(q; n,k) of the rascal-triangle, here q = 3.
%C A306344 The formulas are given for the general case depending on some fixed integer q. The terms are valid for q = 3. For the special case q = 1 see A077028, for q = 2 see A306326. For q < 1 the terms might be negative.
%F A306344 T(q; n,k) = 1 + ((q^k-1)/(q-1))*((q^(n-k)-1)/(q-1)) for 0 <= k <= n.
%F A306344 T(q; n,k) = T(q; n,n-k) for 0 <= k <= n.
%F A306344 T(q; n,0) = T(q; n,n) = 1 for n >= 0.
%F A306344 T(q; n,1) = 1 + (q^(n-1)-1)/(q-1) for n > 0.
%F A306344 T(q; i,j) = 0 if i < j or j < 0.
%F A306344 The T(q; n,k) satisfy several recurrence equations:
%F A306344 (1) T(q; n,k) = q*T(q; n-1,k) + (q^k-1)/(q-1)-(q-1) for 0 <= k < n;
%F A306344 (2) T(q; n,k) = (T(q; n-1,k)*T(q; n-1,k-1) + q^(n-2))/T(q; n-2,k-1),
%F A306344 (3) T(q; n,k) = T(q; n,k-1) + T(q; n-1,k) + q^(n-k-1) - T(q; n-1,k-1),
%F A306344 (4) T(q; n,k) = T(q; n,k-1) + q*T(q; n-2,k-1) - q*T(q; n-2,k-2) for 0 < k < n;
%F A306344 (5) T(q; n,k) = T(q; n,k-2) + T(q; n-1,k) + (1+q)*q^(n-k-1) - T(q; n-1,k-2)
%F A306344 for 1 < k < n with initial values given above.
%F A306344 G.f. of column k >= 0: Sum_{n>=0} T(q; n+k,k)*t^n = (1+((q^k-1)/(q-1)-q)*t) / ((1-t)*(1-q*t)). Take account of lim_{q->1} (q^k-1)/(q-1) = k.
%F A306344 G.f.: Sum_{n>=0, k=0..n} T(q; n,k)*x^k*t^n = (1-q*t-q*x*t+(1+q^2)*x*t^2) / ((1-t)*(1-q*t)*(1-x*t)*(1-q*x*t)).
%F A306344 The row polynomials p(q; n,x) = Sum_{k=0..n} T(q; n,k)*x^k satisfy the recurrence equation p(q; n,x) = q*p(q; n-1,x) + x^n + Sum_{k=0..n-1} ((q^k-1)/(q-1)-(q-1))*x^k for n > 0 with initial value p(q; 0,x) = 1.
%e A306344 If q = 3 the triangle T(3; n,k) starts:
%e A306344 n\k: 0 1 2 3 4 5 6 7 8 9
%e A306344 =============================================================
%e A306344 0: 1
%e A306344 1: 1 1
%e A306344 2: 1 2 1
%e A306344 3: 1 5 5 1
%e A306344 4: 1 14 17 14 1
%e A306344 5: 1 41 53 53 41 1
%e A306344 6: 1 122 161 170 161 122 1
%e A306344 7: 1 365 485 521 521 485 365 1
%e A306344 8: 1 1094 1457 1574 1601 1574 1457 1094 1
%e A306344 9: 1 3281 4373 4733 4841 4841 4733 4373 3281 1
%e A306344 etc.
%Y A306344 Cf. A077028, A306326.
%K A306344 nonn,tabl
%O A306344 0,5
%A A306344 _Werner Schulte_, Feb 08 2019