OFFSET
0,5
COMMENTS
For n >= 1, row n sums to 2*A100314(n).
LINKS
G. C. Greubel, Rows n = 0..100 of the triangle, flattened
FORMULA
From Franck Maminirina Ramaharo, Dec 09 2018:(Start)
n-th row polynomial is 2*(1 - (-1)^(2^n)) + (1 + x)^n + 4*(x - x^n)/(1 - x).
G.f.: (1 - (1 + x)*y + 5*x*y^2 - 4*(x + x^2)*y^3)/((1 - y)*(1 - x*y)*(1 - y - x*y)).
E.g.f.: (4 - 4*x + 4*x*exp(y) - 4*exp(x*y) + (1 - x)*exp((1 + x)*y))/(1 - x). (End)
Sum_{k=0..n} T(n, k) = 2^n + 4*(n - 1 + [n=0]) = 2*A100314(n). - G. C. Greubel, Feb 13 2021
EXAMPLE
Triangle begins:
1;
1, 1;
1, 6, 1;
1, 7, 7, 1;
1, 8, 10, 8, 1;
1, 9, 14, 14, 9, 1;
1, 10, 19, 24, 19, 10, 1;
1, 11, 25, 39, 39, 25, 11, 1;
1, 12, 32, 60, 74, 60, 32, 12, 1;
1, 13, 40, 88, 130, 130, 88, 40, 13, 1;
1, 14, 49, 124, 214, 256, 214, 124, 49, 14, 1;
...
MATHEMATICA
T[n_, m_] = Binomial[n, m] + 4*If[m*(n - m) > 0, 1, 0];
Flatten[Table[T[n, m], {n, 0, 10}, {m, 0, n}]]
PROG
(Maxima) T(n, k) := if k = 0 or k = n then 1 else binomial(n, k) + 4$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Dec 09 2018 */
(Sage)
def T(n, k): return 1 if (k==0 or k==n) else binomial(n, k) + 4
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 13 2021
(Magma)
T:= func< n, k | k eq 0 or k eq n select 1 else Binomial(n, k) + 4 >;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 13 2021
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Feb 23 2010
EXTENSIONS
Edited and name clarified by Franck Maminirina Ramaharo, Dec 09 2018
STATUS
approved