A107711 - OEIS

%I #46 Feb 28 2024 08:10:30

%S 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,

%T 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,

%U 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

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

%C 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.

%C T(2n,n) = A001700(n-1) (n>=1). - _Emeric Deutsch_, Jun 13 2005

%H Reinhard Zumkeller, <a href="/A107711/b107711.txt">Rows n = 1..125 of triangle, flattened</a>

%H Wolfdieter Lang, <a href="http://arxiv.org/abs/1404.2710">On Collatz' Words, Sequences and Trees</a>, arXiv:1404.2710 [math.NT], 2014 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Lang/lang6.html">J. Int. Seq. 17 (2014) # 14.11.7</a>.

%F From _Wolfdieter Lang_, Feb 28 2014 (Start)

%F 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.

%F 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).

%F 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).

%F 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).

%F 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)

%e T(6,2)=5 because binomial(6,2)*gcd(6,2)/6 = 15*2/6 = 5.

%e The triangle T(n,m) begins:

%e n\m 0  1  2   3   4    5   6   7  8  9  10...

%e 0:  1

%e 1:  1  1

%e 2:  1  1  1

%e 3:  1  1  1   1

%e 4:  1  1  3   1   1

%e 5:  1  1  2   2   1    1

%e 6:  1  1  5  10   5    1   1

%e 7:  1  1  3   5   5    3   1   1

%e 8:  1  1  7   7  35    7   7   1  1

%e 9:  1  1  4  28  14   14  28   4  1  1

%e 10: 1  1  9  12  42  126  42  12  9  1   1

%e n\m 0  1  2   3   4    5   6   7  8  9  10...

%e ... reformatted - _Wolfdieter Lang_, Feb 23 2014

%p 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

%t T[0, 0] = 1; T[n_, m_] := Binomial[n, m] * GCD[n, m]/n;

%t Table[T[n, m], {n, 1, 13}, {m, 1, n}] // Flatten (* _Jean-François Alcover_, Nov 16 2017 *)

%o (Haskell)

%o a107711 n k = a107711_tabl !! n !! k

%o a107711_row n = a107711_tabl !! n

%o a107711_tabl = [1] : zipWith (map . flip div) [1..]

%o                (tail $ zipWith (zipWith (*)) a007318_tabl a109004_tabl)

%o -- _Reinhard Zumkeller_, Feb 28 2014

%Y Row sums A159554.

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

%Y Cf. A109004.

%K tabl,nonn

%O 0,13

%A _Leroy Quet_, Jun 10 2005

%E More terms from _Emeric Deutsch_, Jun 13 2005