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A274622 Irregular triangular array read by rows: coefficients in expansion of Gosper's q-sine function sin_q(Pi*z).

%I A274622 #16 Feb 17 2020 21:30:07
%S A274622 1,1,2,2,1,4,4,1,2,8,8,2,4,14,14,4,8,24,24,8,1,14,40,40,14,1,2,24,64,
%T A274622 64,24,2,4,40,100,100,40,4,8,64,154,154,64,8,14,100,232,232,100,14,24,
%U A274622 154,344,344,154,24,1,40,232,504,504,232,40,1,2,64,344,728,728,344,64,2,4,100,504,1040,1040,504,100,4
%N A274622 Irregular triangular array read by rows: coefficients in expansion of Gosper's q-sine function sin_q(Pi*z).
%D A274622 R. W. Gosper, Experiments and discoveries in q-trigonometry, in Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Editors: F. G. Garvan and M. E. H. Ismail. Kluwer, Dordrecht, Netherlands, 2001, pp. 79-105.
%H A274622 R. W. Gosper, <a href="/A274621/a274621.pdf">Experiments and discoveries in q-trigonometry</a>, Preprint.
%H A274622 R. W. Gosper, <a href="/A274621/a274621_1.pdf">q-Trigonometry: Some Prefatory Afterthoughts</a>
%e A274622 The array begins:
%e A274622 .........1,1,
%e A274622 .........2,2,
%e A274622 .......1,4,4,1,
%e A274622 .......2,8,8,2,
%e A274622 .......4,14,14,4,
%e A274622 .......8,24,24,8,
%e A274622 ....1,14,40,40,14,1,
%e A274622 ....2,24,64,64,24,2,
%e A274622 ....4,40,100,100,40,4,
%e A274622 ....8,64,154,154,64,8,
%e A274622 ....14,100,232,232,100,14,
%e A274622 ....24,154,344,344,154,24,
%e A274622 ..1,40,232,504,504,232,40,1,
%e A274622 ..2,64,344,728,728,344,64,2,
%e A274622 ..4,100,504,1040,1040,504,100,4,
%e A274622 ...
%t A274622 nmax = 14; kmax = 4; QP = QPochhammer; s = QP[q^2]/QP[q]^2 + O[q]^(nmax + 1); col[1] = CoefficientList [s, q]; col[k_] := Join[Array[0&, k(k-1)], Take[col[1], nmax-k(k-1)+1]]; T = Transpose[Array[col, kmax]]; ro[n_] := DeleteCases[T[[n+1]], 0]; row[n_] := Join[Reverse[ro[n]], ro[n]]; Table[row[n], {n, 0, nmax}] // Flatten (* _Jean-François Alcover_, Oct 07 2016 *)
%Y A274622 Cf. A274623, A274624, A274625.
%K A274622 nonn,tabf
%O A274622 0,3
%A A274622 _N. J. A. Sloane_, Jul 04 2016