A306344 - OEIS

%I #10 Feb 11 2019 13:28:55

%S 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 122,1,1,365,485,521,521,485,365,1,1,1094,1457,1574,1601,1574,1457,

%U 1094,1,1,3281,4373,4733,4841,4841,4733,4373,3281,1

%N The q-analogs T(q; n,k) of the rascal-triangle, here q = 3.

%C 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 T(q; n,k) = 1 + ((q^k-1)/(q-1))*((q^(n-k)-1)/(q-1)) for 0 <= k <= n.

%F T(q; n,k) = T(q; n,n-k) for 0 <= k <= n.

%F T(q; n,0) = T(q; n,n) = 1 for n >= 0.

%F T(q; n,1) = 1 + (q^(n-1)-1)/(q-1) for n > 0.

%F T(q; i,j) = 0 if i < j or j < 0.

%F The T(q; n,k) satisfy several recurrence equations:

%F   (1) T(q; n,k) = q*T(q; n-1,k) + (q^k-1)/(q-1)-(q-1) for 0 <= k < n;

%F   (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   (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   (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   (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   for 1 < k < n  with initial values given above.

%F 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 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 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 If q = 3 the triangle T(3; n,k) starts:

%e n\k:  0     1     2     3     4     5     6     7     8     9

%e =============================================================

%e   0:  1

%e   1:  1     1

%e   2:  1     2     1

%e   3:  1     5     5     1

%e   4:  1    14    17    14     1

%e   5:  1    41    53    53    41     1

%e   6:  1   122   161   170   161   122     1

%e   7:  1   365   485   521   521   485   365     1

%e   8:  1  1094  1457  1574  1601  1574  1457  1094     1

%e   9:  1  3281  4373  4733  4841  4841  4733  4373  3281     1

%e etc.

%Y Cf. A077028, A306326.

%K nonn,tabl

%O 0,5

%A _Werner Schulte_, Feb 08 2019