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.

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, 36, 115, 215, 261, 215, 121, 45, 10, 1, 0, 6, 52, 206, 495, 806, 938, 798, 496, 220, 66, 12, 1, 0, 7, 71, 336, 987, 2016, 3031, 3452, 3010, 2003, 1001, 364, 91

Offset: 0

Author

Franck Maminirina Ramaharo, Feb 28 2018

Keywords

• nonn
• tabf

Examples

			The triangle T(n, k) begins:
n\k  0  1   2    3    4     5     6     7     8     9    10   11  12  13 14
0:   0  0   1
1:   0  1   2    1
2:   0  2   6    6    2
3:   0  3  13   22   18     7     1
4:   0  4  23   56   75    60    29     8     1
5:   0  5  36  115  215   261   215   121    45    10     1
6:   0  6  52  206  495   806   938   798   496   220    66   12   1
7:   0  7  71  336  987  2016  3031  3452  3010  2003  1001  364  91  14  1
		

References

• M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
• J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996.

Links

• Paul Barry, On the Connection Coefficients of the Chebyshev-Boubaker polynomials, The Scientific World Journal, Volume 2013 (2013), Article ID 657806, 10 pages.
• Milan Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013.
• A. M. Mathai and P. N. Rathie, Enumeration of almost cubic maps, Journal of Combinatorial Theory, Series B, Vol 13 (1972), 83-90.
• Franck Ramaharo, A one-variable bracket polynomial for some Turk's head knots, arXiv:1807.05256 [math.CO], 2018.
• Franck Ramaharo, A bracket polynomial for 2-tangle shadows, arXiv:2002.06672 [math.CO], 2020.

Crossrefs

Row sums: A000302 (powers of 4).

Cf. A034870, A034871, A032443.

Programs

Maple

T := (n, k) -> binomial(2*n, k) + binomial(n, k - 2) - binomial(n, k);
for n from 0 to 10 do seq(T(n, k), k = 0 .. max(2*n, n + 2)) od;

Maxima

T(n, k) := binomial(2*n, k) + binomial(n, k - 2) - binomial(n, k)$
a : []$
for n:0 thru 10 do
  a : append(a, makelist(T(n, k), k, 0, max(2*n, n + 2)))$
a;

PARI

row(n) = Vecrev((x^2 + 2*x + 1)^n + (x^2 - 1)*(x + 1)^n); \\ Michel Marcus, Nov 12 2022

Formula

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

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

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

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

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

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

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

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

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

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