A300192 - OEIS

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 #36 Nov 12 2022 05:52:53

%S 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 36,115,215,261,215,121,45,10,1,0,6,52,206,495,806,938,798,496,220,66,

%U 12,1,0,7,71,336,987,2016,3031,3452,3010,2003,1001,364,91

%N 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 M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.

%D J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996.

%H 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 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 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 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 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 T(n,k) = binomial(2*n,k) + binomial(n,k-2) - binomial(n,k).

%F 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 T(n,1) = A001477(n).

%F T(n,2) = A143689(n).

%F T(n,3) = n + A002492(n-1) - A000292(n-2).

%F T(n,n) = A247493(n+1,n).

%F T(n,n+1) = n + A001791(n).

%F T(n,n+2) = 1 + A002694(n), n >= 2.

%F T(n,n+k) = binomial(2*n, n-k) = A094527(n,k), for k >= 3 and n>=k.

%F G.f.: 1/(1 - y*(x^2 + 2*x + 1)) + (x^2 - 1)/(1 - y*(x + 1)).

%e The triangle T(n, k) begins:

%e n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

%e 0: 0 0 1

%e 1: 0 1 2 1

%e 2: 0 2 6 6 2

%e 3: 0 3 13 22 18 7 1

%e 4: 0 4 23 56 75 60 29 8 1

%e 5: 0 5 36 115 215 261 215 121 45 10 1

%e 6: 0 6 52 206 495 806 938 798 496 220 66 12 1

%e 7: 0 7 71 336 987 2016 3031 3452 3010 2003 1001 364 91 14 1

%p T := (n, k) -> binomial(2*n, k) + binomial(n, k - 2) - binomial(n, k);

%p for n from 0 to 10 do seq(T(n, k), k = 0 .. max(2*n, n + 2)) od;

%o (Maxima)

%o T(n, k) := binomial(2*n, k) + binomial(n, k - 2) - binomial(n, k)$

%o a : []$

%o for n:0 thru 10 do

%o a : append(a, makelist(T(n, k), k, 0, max(2*n, n + 2)))$

%o a;

%o (PARI) row(n) = Vecrev((x^2 + 2*x + 1)^n + (x^2 - 1)*(x + 1)^n); \\ _Michel Marcus_, Nov 12 2022

%Y Row sums: A000302 (powers of 4).

%Y Cf. A034870, A034871, A032443.

%K nonn,tabf

%O 0,6

%A _Franck Maminirina Ramaharo_, Feb 28 2018