OFFSET
0,6
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.
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,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)).
EXAMPLE
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
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;
PROG
(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
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Franck Maminirina Ramaharo, Feb 28 2018
STATUS
approved