A107711 Triangle read by rows: T(0,0)=1, T(n,m) = binomial(n,m) * gcd(n,m)/n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 5, 10, 5, 1, 1, 1, 1, 3, 5, 5, 3, 1, 1, 1, 1, 7, 7, 35, 7, 7, 1, 1, 1, 1, 4, 28, 14, 14, 28, 4, 1, 1, 1, 1, 9, 12, 42, 126, 42, 12, 9, 1, 1, 1, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1, 1, 1, 11, 55, 165, 66, 462, 66, 165, 55, 11, 1, 1

Offset: 0

Author

Leroy Quet, Jun 10 2005

Keywords

• tabl
• nonn

Comments

T(0,0) is an indeterminate, but 1 seems a logical value to assign it. T(n,0) = T(n,1) = T(n,n-1) = T(n,n) = 1.

T(2n,n) = A001700(n-1) (n>=1). - Emeric Deutsch, Jun 13 2005

Examples

			T(6,2)=5 because binomial(6,2)*gcd(6,2)/6 = 15*2/6 = 5.
The triangle T(n,m) begins:
n\m 0  1  2   3   4    5   6   7  8  9  10...
0:  1
1:  1  1
2:  1  1  1
3:  1  1  1   1
4:  1  1  3   1   1
5:  1  1  2   2   1    1
6:  1  1  5  10   5    1   1
7:  1  1  3   5   5    3   1   1
8:  1  1  7   7  35    7   7   1  1
9:  1  1  4  28  14   14  28   4  1  1
10: 1  1  9  12  42  126  42  12  9  1   1
n\m 0  1  2   3   4    5   6   7  8  9  10...
... reformatted - _Wolfdieter Lang_, Feb 23 2014
		

Links

• Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
• Wolfdieter Lang, On Collatz' Words, Sequences and Trees, arXiv:1404.2710 [math.NT], 2014 and J. Int. Seq. 17 (2014) # 14.11.7.

Crossrefs

Row sums A159554.

Cf. A007318, A001700, A000108, A050169, A234040, A026741, A234042.

Cf. A109004.

Programs

Haskell

a107711 n k = a107711_tabl !! n !! k
a107711_row n = a107711_tabl !! n
a107711_tabl = [1] : zipWith (map . flip div) [1..]
               (tail $ zipWith (zipWith (*)) a007318_tabl a109004_tabl)
-- Reinhard Zumkeller, Feb 28 2014

Maple

a:=proc(n,k) if n=0 and k=0 then 1 elif k<=n then binomial(n,k)*gcd(n,k)/n else 0 fi end: for n from 0 to 13 do seq(a(n,k),k=0..n) od; # yields sequence in triangular form. - Emeric Deutsch, Jun 13 2005

Mathematica

T[0, 0] = 1; T[n_, m_] := Binomial[n, m] * GCD[n, m]/n;
Table[T[n, m], {n, 1, 13}, {m, 1, n}] // Flatten (* Jean-François Alcover, Nov 16 2017 *)

Formula

From Wolfdieter Lang, Feb 28 2014 (Start)

T(n, m) = T(n-1,m)*(n-1)*gcd(n,m)/((n-m)*gcd(n-1,m)), n > m >= 1, T(n, 0) = 1, T(n, n) = 1, otherwise 0.

T(n, m) = binomial(n-1,m-1)*gcd(n,m)/m for n >= m >= 1, T(n,0) = 1, otherwise 0 (from iteration of the preceding recurrence).

T(n, m) = T(n-1, m-1)*(n-1)*gcd(n,m)/(m*gcd(n-1,m-1)) for n >= m >= 2, T(n, 0) = 1, T(n, 1) = 0, otherwise 0 (from the preceding formula).

T(2*n, n) = A001700(n-1) (n>=1) (see the Emeric Deutsch comment above), T(2*n, n-1) = A234040(n), T(2*n+1,n) = A000108(n), n >= 0 (Catalan numbers).

Column sequences: T(n+2, 2) = A026741(n+1), T(n+3, 3) = A234041(n), T(n+4, 4) = A208950(n+2), T(n+5, 5) = A234042, n >= 0. (End)

Extensions

More terms from Emeric Deutsch, Jun 13 2005