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

0,5

Comments

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.

Links

Table of n, a(n) for n=0..54.

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.

Example

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.

Crossrefs

Cf. A077028, A306326.

Sequence in context: A099927 A139332 A187617 * A128612 A284731 A211400

Adjacent sequences: A306341 A306342 A306343 * A306345 A306346 A306347

Keyword

nonn,tabl

Author

Werner Schulte, Feb 08 2019

Status

approved