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A107711
Triangle read by rows: T(0,0)=1, T(n,m) = binomial(n,m) * gcd(n,m)/n.
9
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
0,13
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
LINKS
Wolfdieter Lang, On Collatz' Words, Sequences and Trees, arXiv:1404.2710 [math.NT], 2014 and J. Int. Seq. 17 (2014) # 14.11.7.
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)
EXAMPLE
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
tabl,nonn
AUTHOR
Leroy Quet, Jun 10 2005
EXTENSIONS
More terms from Emeric Deutsch, Jun 13 2005
STATUS
approved