A306344 The q-analogs T(q; n,k) of the rascal-triangle, here q = 3.
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, 122, 1, 1, 365, 485, 521, 521, 485, 365, 1, 1, 1094, 1457, 1574, 1601, 1574, 1457, 1094, 1, 1, 3281, 4373, 4733, 4841, 4841, 4733, 4373, 3281, 1
Offset: 0
Examples
If q = 3 the triangle T(3; n,k) starts: n\k: 0 1 2 3 4 5 6 7 8 9 ============================================================= 0: 1 1: 1 1 2: 1 2 1 3: 1 5 5 1 4: 1 14 17 14 1 5: 1 41 53 53 41 1 6: 1 122 161 170 161 122 1 7: 1 365 485 521 521 485 365 1 8: 1 1094 1457 1574 1601 1574 1457 1094 1 9: 1 3281 4373 4733 4841 4841 4733 4373 3281 1 etc.
Formula
T(q; n,k) = 1 + ((q^k-1)/(q-1))*((q^(n-k)-1)/(q-1)) for 0 <= k <= n.
T(q; n,k) = T(q; n,n-k) for 0 <= k <= n.
T(q; n,0) = T(q; n,n) = 1 for n >= 0.
T(q; n,1) = 1 + (q^(n-1)-1)/(q-1) for n > 0.
T(q; i,j) = 0 if i < j or j < 0.
The T(q; n,k) satisfy several recurrence equations:
(1) T(q; n,k) = q*T(q; n-1,k) + (q^k-1)/(q-1)-(q-1) for 0 <= k < n;
(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),
(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),
(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;
(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)
for 1 < k < n with initial values given above.
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.
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)).
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.
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