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%I #18 Feb 11 2019 13:28:46
%S 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,
%T 64,94,106,106,94,64,1,1,128,190,218,226,218,190,128,1,1,256,382,442,
%U 466,466,442,382,256,1,1,512,766,890,946,962,946,890,766,512,1
%N The q-analogs T(q; n,k) of the rascal-triangle, here q = 2.
%C 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.
%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 = 2 the triangle T(2; 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 4 4 1
%e 4: 1 8 10 8 1
%e 5: 1 16 22 22 16 1
%e 6: 1 32 46 50 46 32 1
%e 7: 1 64 94 106 106 94 64 1
%e 8: 1 128 190 218 226 218 190 128 1
%e 9: 1 256 382 442 466 466 442 382 256 1
%e etc.
%Y Cf. A077028, A306344.
%K nonn,tabl
%O 0,5
%A _Werner Schulte_, Feb 07 2019
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