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A173740
Triangle T(n,k) = binomial(n,k) + 2 for 1 <= k <= n - 1, n >= 2, and T(n,0) = T(n,n) = 1 for n >= 0, read by rows.
5
1, 1, 1, 1, 4, 1, 1, 5, 5, 1, 1, 6, 8, 6, 1, 1, 7, 12, 12, 7, 1, 1, 8, 17, 22, 17, 8, 1, 1, 9, 23, 37, 37, 23, 9, 1, 1, 10, 30, 58, 72, 58, 30, 10, 1, 1, 11, 38, 86, 128, 128, 86, 38, 11, 1, 1, 12, 47, 122, 212, 254, 212, 122, 47, 12, 1, 1, 13, 57, 167, 332, 464, 464, 332, 167, 57, 13, 1
OFFSET
0,5
COMMENTS
For n >= 1, row n sums to A131520(n).
FORMULA
From Franck Maminirina Ramaharo, Dec 08 2018:(Start)
T(n,k) = A007318(n,k) + 2*(1 - A103451(n,k)).
T(n,k) = 3*A007318(n,k) - 2*A132044(n,k).
n-th row polynomial is 1 - (-1)^(2^n) + (1 + x)^n + 2*(x - x^n)/(1 - x).
G.f.: (1 - (1 + x)*y + 3*x*y^2 - 2*(x + x^2)*y^3)/((1 - y)*(1 - x*y)*(1 - y - x*y)).
E.g.f.: (2 - 2*x + 2*x*exp(y) - 2*exp(x*y) + (1 - x)*exp((1 + x)*y))/(1 - x). (End)
Sum_{k=0..n} T(n, k) = 2^n + 2*(n - 1 + [n=0]) = 2*A100314(n). - G. C. Greubel, Feb 13 2021
EXAMPLE
Triangle begins:
1;
1, 1;
1, 4, 1;
1, 5, 5, 1;
1, 6, 8, 6, 1;
1, 7, 12, 12, 7, 1;
1, 8, 17, 22, 17, 8, 1;
1, 9, 23, 37, 37, 23, 9, 1;
1, 10, 30, 58, 72, 58, 30, 10, 1;
1, 11, 38, 86, 128, 128, 86, 38, 11, 1;
1, 12, 47, 122, 212, 254, 212, 122, 47, 12, 1;
...
MATHEMATICA
T[n_, m_] = Binomial[n, m] + 2*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) + 2$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Dec 08 2018 */
(Sage)
def T(n, k): return 1 if (k==0 or k==n) else binomial(n, k) + 2
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) + 2 >;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 13 2021
CROSSREFS
Sequences of the form binomial(n, k) + q: A132823 (q=-2), A132044 (q=-1), A007318 (q=0), A132735 (q=1), this sequence (q=2), A173741 (q=4), A173742 (q=6).
Sequence in context: A153843 A318795 A099575 * A028275 A173118 A147289
KEYWORD
nonn,tabl,easy
AUTHOR
Roger L. Bagula, Feb 23 2010
EXTENSIONS
Edited and name clarified by Franck Maminirina Ramaharo, Dec 08 2018
STATUS
approved