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A306326
The q-analogs T(q; n,k) of the rascal-triangle, here q = 2.
1
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
0,5
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.
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 = 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
Sequence in context: A056588 A126770 A202979 * A156006 A137854 A062715
KEYWORD
nonn,tabl
AUTHOR
Werner Schulte, Feb 07 2019
STATUS
approved