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A300192 Triangle read by rows: row n consists of the coefficients of the expansion of the polynomial (x^2 + 2*x + 1)^n + (x^2 - 1)*(x + 1)^n.

%I A300192 #36 Nov 12 2022 05:52:53
%S A300192 0,0,1,0,1,2,1,0,2,6,6,2,0,3,13,22,18,7,1,0,4,23,56,75,60,29,8,1,0,5,
%T A300192 36,115,215,261,215,121,45,10,1,0,6,52,206,495,806,938,798,496,220,66,
%U A300192 12,1,0,7,71,336,987,2016,3031,3452,3010,2003,1001,364,91
%N A300192 Triangle read by rows: row n consists of the coefficients of the expansion of the polynomial (x^2 + 2*x + 1)^n + (x^2 - 1)*(x + 1)^n.
%D A300192 M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
%D A300192 J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996.
%H A300192 Paul Barry, <a href="http://dx.doi.org/10.1155/2013/657806">On the Connection Coefficients of the Chebyshev-Boubaker polynomials</a>, The Scientific World Journal, Volume 2013 (2013), Article ID 657806, 10 pages.
%H A300192 Milan Janjic and B. Petkovic, <a href="https://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv preprint arXiv:1301.4550 [math.CO], 2013.
%H A300192 A. M. Mathai and  P. N. Rathie, <a href="https://doi.org/10.1016/0095-8956(72)90012-3">Enumeration of almost cubic maps</a>, Journal of Combinatorial Theory, Series B, Vol 13 (1972), 83-90.
%H A300192 Franck Ramaharo, <a href="https://arxiv.org/abs/1807.05256">A one-variable bracket polynomial for some Turk's head knots</a>, arXiv:1807.05256 [math.CO], 2018.
%H A300192 Franck Ramaharo, <a href="https://arxiv.org/abs/2002.06672">A bracket polynomial for 2-tangle shadows</a>, arXiv:2002.06672 [math.CO], 2020.
%F A300192 T(n,k) = binomial(2*n,k) + binomial(n,k-2) - binomial(n,k).
%F A300192 T(n,k) = T(n-1,k-1)+ T(n-1,k) + A034871(n-1,k-1), with T(n,0) = T(0,1) = 0 and T(0,2) = 1
%F A300192 T(n,1) = A001477(n).
%F A300192 T(n,2) = A143689(n).
%F A300192 T(n,3) = n + A002492(n-1) - A000292(n-2).
%F A300192 T(n,n) = A247493(n+1,n).
%F A300192 T(n,n+1) = n + A001791(n).
%F A300192 T(n,n+2) = 1 + A002694(n), n >= 2.
%F A300192 T(n,n+k) = binomial(2*n, n-k) = A094527(n,k), for k >= 3 and n>=k.
%F A300192 G.f.: 1/(1 - y*(x^2 + 2*x + 1)) + (x^2 - 1)/(1 - y*(x + 1)).
%e A300192 The triangle T(n, k) begins:
%e A300192 n\k  0  1   2    3    4     5     6     7     8     9    10   11  12  13 14
%e A300192 0:   0  0   1
%e A300192 1:   0  1   2    1
%e A300192 2:   0  2   6    6    2
%e A300192 3:   0  3  13   22   18     7     1
%e A300192 4:   0  4  23   56   75    60    29     8     1
%e A300192 5:   0  5  36  115  215   261   215   121    45    10     1
%e A300192 6:   0  6  52  206  495   806   938   798   496   220    66   12   1
%e A300192 7:   0  7  71  336  987  2016  3031  3452  3010  2003  1001  364  91  14  1
%p A300192 T := (n, k) -> binomial(2*n, k) + binomial(n, k - 2) - binomial(n, k);
%p A300192 for n from 0 to 10 do seq(T(n, k), k = 0 .. max(2*n, n + 2)) od;
%o A300192 (Maxima)
%o A300192 T(n, k) := binomial(2*n, k) + binomial(n, k - 2) - binomial(n, k)$
%o A300192 a : []$
%o A300192 for n:0 thru 10 do
%o A300192   a : append(a, makelist(T(n, k), k, 0, max(2*n, n + 2)))$
%o A300192 a;
%o A300192 (PARI) row(n) = Vecrev((x^2 + 2*x + 1)^n + (x^2 - 1)*(x + 1)^n); \\ _Michel Marcus_, Nov 12 2022
%Y A300192 Row sums: A000302 (powers of 4).
%Y A300192 Cf. A034870, A034871, A032443.
%K A300192 nonn,tabf
%O A300192 0,6
%A A300192 _Franck Maminirina Ramaharo_, Feb 28 2018