A306326 The q-analogs T(q; n,k) of the rascal-triangle, here q = 2.

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 10, 8, 1, 1, 16, 22, 22, 16, 1, 1, 32, 46, 50, 46, 32, 1, 1, 64, 94, 106, 106, 94, 64, 1, 1, 128, 190, 218, 226, 218, 190, 128, 1, 1, 256, 382, 442, 466, 466, 442, 382, 256, 1, 1, 512, 766, 890, 946, 962, 946, 890, 766, 512, 1

Offset: 0

Author

Werner Schulte, Feb 07 2019

Keywords

• nonn
• tabl

Comments

The formulas are given for the general case depending on some fixed integer q. The terms are valid if q = 2. For the special case q = 1 see A077028, for q = 3 see A306344. For q < 1 the terms might be negative.

Examples

			If q = 2 the triangle T(2; 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     4     4     1
  4:  1     8    10     8     1
  5:  1    16    22    22    16     1
  6:  1    32    46    50    46    32     1
  7:  1    64    94   106   106    94    64     1
  8:  1   128   190   218   226   218   190   128     1
  9:  1   256   382   442   466   466   442   382   256     1
etc.
		

Crossrefs

Cf. A077028, A306344.

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.