A159554 -id:A159554 - cp's OEIS frontend

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

Leroy Quet, Jun 10 2005

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

			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

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.

Row sums A159554.
Cf. A007318, A001700, A000108, A050169, A234040, A026741, A234042.
Cf. A109004.

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

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[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 *)

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)

More terms from Emeric Deutsch, Jun 13 2005

A159553 a(n) = Sum_{k=0..n} binomial(n,k) * gcd(n,k).

Original entry on oeis.org

2, 6, 12, 28, 40, 144, 140, 536, 864, 2560, 2068, 12720, 8216, 45192, 78660, 182832, 131104, 933984, 524324, 3698240, 4890648, 13345816, 8388652, 67390464, 60129600, 225470544, 279938160, 1032462256, 536870968, 5018059200

Offset: 1

Leroy Quet, Apr 14 2009

nonn

For the purpose of this sequence, gcd(n,0) = n, for all positive integers n.

a(n) is a multiple of n, for all nonnegative integers n.

Michael De Vlieger, Table of n, a(n) for n = 1..3317

Cf. A159068, A159554.

A159553 := proc(n) add(binomial(n, k)*gcd(k, n), k=0..n) ; end: seq(A159553(n),n=1..40) ; # R. J. Mathar, Apr 29 2009

Table[Sum[Binomial[n, k] GCD[n, k], {k, 0, n}], {n, 30}] (* Michael De Vlieger, Oct 30 2017 *)

a(n) = A159068(n) + n.

a(n) = 2^n * Sum_{d|n} (phi(d)/d) * Sum_{k=1..d} (-1)^(k*n/d)*cos(k*Pi/d)^n.

Extended by R. J. Mathar, Apr 29 2009

Ambiguous term a(0) removed by Max Alekseyev, Jan 09 2015

A159555 Numbers m where m^2 divides A159553(m), where A159553(m) = Sum_{k=0..m} binomial(m,k) * gcd(m,k).

Original entry on oeis.org

1, 6, 22, 72, 114, 148, 164, 260, 261, 780, 1078, 1184, 1266, 2952, 4674, 21868

Offset: 1

Leroy Quet, Apr 15 2009

For the purpose of this sequence, gcd(m,0) = m.

No other term up to 15000. - Michel Marcus, Sep 06 2019

Cf. A159458, A159553, A159554.

A159068 := proc(n) option remember; add(binomial(n, k)*gcd(k, n), k=1..n) ; end: A159553 := proc(n) option remember ; A159068(n)+n; end: isA159555 := proc(n) if A159553(n) mod ( n^2) = 0 then true; else false; fi; end: for n from 1 do if isA159555(n) then printf("%d,\n",n) ; fi; od: # R. J. Mathar, Apr 29 2009

f(n) = sum(k=0, n, binomial(n,k) * gcd(n,k)); \\ A159553
isok(n) = !(f(n) % n^2); \\ Michel Marcus, Sep 05 2019

Extended by R. J. Mathar, Apr 29 2009
a(14)-a(15) from Ray Chandler, Jun 18 2009
a(16) from Jinyuan Wang, Jul 25 2022

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

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A159553 a(n) = Sum_{k=0..n} binomial(n,k) * gcd(n,k).

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A159555 Numbers m where m^2 divides A159553(m), where A159553(m) = Sum_{k=0..m} binomial(m,k) * gcd(m,k).

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