OFFSET
0,8
COMMENTS
Starting 1,0,1,1,1,... this is the Riordan array ((1-x+x^2)/(1-x), x/(1-x)). Its diagonal sums are A006355. Its inverse is A106509. - Paul Barry, May 04 2005
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
FORMULA
T(n, k) = C(n,k) - C(n-2,k-1) for n >= 3 and T(n, k) = 1 otherwise.
T(n, k) = T(n-1, k-1) + T(n-1, k) starting with T(2, 0) = T(2, 1) = T(2, 2) = 1 and T(n, 0) = T(n, n) = 1.
G.f.: (1-x^2*y) / (1 - x*(1+y)). - Ralf Stephan, Jan 31 2005
From G. C. Greubel, Apr 28 2021: (Start)
Sum_{k=0..n} T(n, k) = (n+1)*[n<3] + 3*2^(n-2)*[n>=3].
T(n, k, q) = q*[n=2] + Sum_{j=0..5} q^j*binomial(n-2*j, k-j)*[n>2*j] with T(n,0) = T(n,n) = 1 for q = -1. (End)
EXAMPLE
Rows start as:
1;
1, 1;
1, 1, 1; (key row for starting the recurrence)
1, 2, 2, 1;
1, 3, 4, 3, 1;
1, 4, 7, 7, 4, 1;
1, 5, 11, 14, 11, 5, 1;
MATHEMATICA
t[2, 1] = 1; t[n_, n_] = t[_, 0] = 1; t[n_, k_] := t[n, k] = t[n-1, k-1] + t[n-1, k]; Table[t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 28 2013, after Ralf Stephan *)
PROG
(Magma)
T:= func< n, k | n lt 3 select 1 else Binomial(n, k) - Binomial(n-2, k-1) >;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 28 2021
(Sage)
def T(n, k): return 1 if n<3 else binomial(n, k) - binomial(n-2, k-1)
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 28 2021
(PARI) A072405(n, k) = if(n>2, binomial(n, k)-binomial(n-2, k-1), 1) \\ M. F. Hasler, Jan 06 2024
CROSSREFS
Cf. A072406 for number of odd terms in each row.
KEYWORD
AUTHOR
Henry Bottomley, Jun 16 2002
STATUS
approved