A072405 Triangle T(n, k) = C(n,k) - C(n-2,k-1) for n >= 3 and T(n, k) = 1 otherwise, read by rows.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 4, 3, 1, 1, 4, 7, 7, 4, 1, 1, 5, 11, 14, 11, 5, 1, 1, 6, 16, 25, 25, 16, 6, 1, 1, 7, 22, 41, 50, 41, 22, 7, 1, 1, 8, 29, 63, 91, 91, 63, 29, 8, 1, 1, 9, 37, 92, 154, 182, 154, 92, 37, 9, 1, 1, 10, 46, 129, 246, 336, 336, 246, 129, 46, 10, 1, 1, 11, 56, 175, 375, 582, 672, 582, 375, 175, 56, 11, 1

Offset: 0

Author

Henry Bottomley, Jun 16 2002

Keywords

• easy
• nonn
• tabl

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

Examples

			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;
		

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.

Crossrefs

Row sums give essentially A003945, A007283, or A042950.

Cf. A072406 for number of odd terms in each row.

Cf. A051597, A096646, A122218 (identical for n > 1).

Cf. A007318 (q=0), A072405 (q= -1), A173117 (q=1), A173118 (q=2), A173119 (q=3), A173120 (q=-4).

Programs

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

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 *)

PARI

A072405(n, k) = if(n>2, binomial(n, k)-binomial(n-2, k-1), 1) \\ M. F. Hasler, Jan 06 2024

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

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)