A004736 -id:A004736 - cp's OEIS frontend

A036561 Nicomachus triangle read by rows, T(n, k) = 2^(n - k)*3^k, for 0 <= k <= n.

1, 2, 3, 4, 6, 9, 8, 12, 18, 27, 16, 24, 36, 54, 81, 32, 48, 72, 108, 162, 243, 64, 96, 144, 216, 324, 486, 729, 128, 192, 288, 432, 648, 972, 1458, 2187, 256, 384, 576, 864, 1296, 1944, 2916, 4374, 6561, 512, 768, 1152, 1728, 2592, 3888, 5832, 8748, 13122, 19683

Offset: 0

N. J. A. Sloane

The triangle pertaining to this sequence has the property that every row, every column and every diagonal contains a nontrivial geometric progression. More interestingly every line joining any two elements contains a nontrivial geometric progression. - Amarnath Murthy, Jan 02 2002
Kappraff states (pp. 148-149): "I shall refer to this as Nicomachus' table since an identical table of numbers appeared in the Arithmetic of Nicomachus of Gerasa (circa 150 A.D.)" The table was rediscovered during the Italian Renaissance by Leon Battista Alberti, who incorporated the numbers in dimensions of his buildings and in a system of musical proportions. Kappraff states "Therefore a room could exhibit a 4:6 or 6:9 ratio but not 4:9. This ensured that ratios of these lengths would embody musical ratios". - Gary W. Adamson, Aug 18 2003
After Nichomachus and Alberti several Renaissance authors described this table. See for instance Pierre de la Ramée in 1569 (facsimile of a page of his Arithmetic Treatise in Latin in the links section). - Olivier Gérard, Jul 04 2013
The triangle sums, see A180662 for their definitions, link Nicomachus's table with eleven different sequences, see the crossrefs. It is remarkable that these eleven sequences can be described with simple elegant formulas. The mirror of this triangle is A175840. - Johannes W. Meijer, Sep 22 2010
The diagonal sums Sum_{k} T(n - k, k) give A167762(n + 2). - Michael Somos, May 28 2012
Where d(n) is the divisor count function, then d(T(i,j)) = A003991, the rows of which sum to the tetrahedral numbers A000292(n+1). For example, the sum of the divisors of row 4 of this triangle (i = 4), gives d(16) + d(24) + d(36) + d(54) + d(81) = 5 + 8 + 9 + 8 + 5 = 35 = A000292(5). In fact, where p and q are distinct primes, the aforementioned relationship to the divisor function and tetrahedral numbers can be extended to any triangle of numbers in which the i-th row is of form {p^(i-j)*q^j, 0<=j<=i}; i >= 0 (e.g., A003593, A003595). - Raphie Frank, Nov 18 2012, corrected Dec 07 2012
Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then 2*x and 3*x are in S, and duplicates are deleted as they occur; see A232559. - Clark Kimberling, Nov 28 2013
Partial sums of rows produce Stirling numbers of the 2nd kind: A000392(n+2) = Sum_{m=1..(n^2+n)/2} a(m). - Fred Daniel Kline, Sep 22 2014
A permutation of A003586. - L. Edson Jeffery, Sep 22 2014
Form a word of length i by choosing a (possibly empty) word on alphabet {0,1} then concatenating a word of length j on alphabet {2,3,4}. T(i,j) is the number of such words. - Geoffrey Critzer, Jun 23 2016
Form of Zorach additive triangle (see A035312) where each number is sum of west and northwest numbers, with the additional condition that each number is GCD of the two numbers immediately below it. - Michel Lagneau, Dec 27 2018

			The start of the sequence as a triangular array read by rows:
   1
   2   3
   4   6   9
   8  12  18  27
  16  24  36  54  81
  32  48  72 108 162 243
  ...
The start of the sequence as a table T(n,k) n, k > 0:
    1    2    4    8   16   32 ...
    3    6   12   24   48   96 ...
    9   18   36   72  144  288 ...
   27   54  108  216  432  864 ...
   81  162  324  648 1296 2592 ...
  243  486  972 1944 3888 7776 ...
  ...
- _Boris Putievskiy_, Jan 08 2013

Jay Kappraff, Beyond Measure, World Scientific, 2002, p. 148.
Flora R. Levin, The Manual of Harmonics of Nicomachus the Pythagorean, Phanes Press, 1994, p. 114.

Reinhard Zumkeller and Matthew House, Rows n = 0..300 of triangle, flattened [Rows 0 through 120 were computed by Reinhard Zumkeller; rows 121 through 300 by Matthew House, Jul 09 2015]
Fred Daniel Kline, How do I convert this Nicomachus' Triangle to one with edges?
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Pierre de la Ramée (Petrus Ramus), P. Rami Arithmeticae (anno 1569) Liber 2, Cap. XVI "De inventione continue proportionalium" p.46 (leaf 0055) describes this integer triangle in a layout close to the current OEIS 'tabl' layout.
Marko Riedel, Proof of identity by Egorychev method.
Thomas Scheuerle, Version of this triangle from Boethius (480-524), Anicius Manlius Severinus Boethius, De institutione arithmetica, Medeltidshandskrift 1 (Mh 1), Lund University Library, early 10th century, page 4r.
Robert Sedgewick, Analysis of shellsort and related algorithms, Fourth European Symposium on Algorithms, Barcelona, September, 1996.

Cf. A001047 (row sums), A000400 (central terms), A013620, A007318.
Cf. A003586, A000079, A000244, A007283, A025197, A005010, A003946, A005051.
Triangle sums (see the comments): A001047 (Row1); A015441 (Row2); A005061 (Kn1, Kn4); A016133 (Kn2, Kn3); A016153 (Fi1, Fi2); A016140 (Ca1, Ca4); A180844 (Ca2, Ca3); A180845 (Gi1, Gi4); A180846 (Gi2, Gi3); A180847 (Ze1, Ze4); A016185 (Ze2, Ze3). - Johannes W. Meijer, Sep 22 2010, Sep 10 2011
Antidiagonal cumulative sum: A000392; square arrays cumulative sum: A160869. Antidiagonal products: 6^A000217; antidiagonal cumulative products: 6^A000292; square arrays products: 6^A005449; square array cumulative products: 6^A006002.

a036561 n k = a036561_tabf !! n !! k
a036561_row n = a036561_tabf !! n
a036561_tabf = iterate (\xs@(x:_) -> x * 2 : map (* 3) xs) [1]
-- Reinhard Zumkeller, Jun 08 2013

/* As triangle: */ [[(2^(i-j)*3^j)/3: j in [1..i]]: i in [1..10]]; // Vincenzo Librandi, Oct 17 2014

A036561 := proc(n,k): 2^(n-k)*3^k end:
seq(seq(A036561(n,k),k=0..n),n=0..9);
T := proc(n,k) option remember: if k=0 then 2^n elif k>=1 then procname(n,k-1) + procname(n-1,k-1) fi: end: seq(seq(T(n,k),k=0..n),n=0..9);
# Johannes W. Meijer, Sep 22 2010, Sep 10 2011

Flatten[Table[ 2^(i-j) 3^j, {i, 0, 12}, {j, 0, i} ]] (* Flatten added by Harvey P. Dale, Jun 07 2011 *)

for(i=0,9,for(j=0,i,print1(3^j<<(i-j)", "))) \\ Charles R Greathouse IV, Dec 22 2011

{T(n, k) = if( k<0 || k>n, 0, 2^(n - k) * 3^k)} /* Michael Somos, May 28 2012 */

T(n,k) = A013620(n,k)/A007318(n,k). - Reinhard Zumkeller, May 14 2006
T(n,k) = T(n,k-1) + T(n-1,k-1) for n>=1 and 1<=k<=n with T(n,0) = 2^n for n>=0. - Johannes W. Meijer, Sep 22 2010
T(n,k) = 2^(k-1)*3^(n-1), n, k > 0 read by antidiagonals. - Boris Putievskiy, Jan 08 2013
a(n) = 2^(A004736(n)-1)*3^(A002260(n)-1), n > 0, or a(n) = 2^(j-1)*3^(i-1) n > 0, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor[(-1+sqrt(8*n-7))/2]. - Boris Putievskiy, Jan 08 2013
G.f.: 1/((1-2x)(1-3yx)). - Geoffrey Critzer, Jun 23 2016
T(n,k) = (-1)^n * Sum_{q=0..n} (-1)^q * C(k+3*q, q) * C(n+2*q, n-q). - Marko Riedel, Jul 01 2024

A286561 Square array A(n,k): A(n,1) = 1, and for k > 1, A(n,k) = the highest exponent e such that k^e divides n, read by descending antidiagonals as A(1,1), A(1,2), A(2,1), etc.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1

Offset: 1

Antti Karttunen, May 20 2017

			The top left 18 X 18 corner of the array:
  n \k 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18
     .-----------------------------------------------------
   1 | 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
   2 | 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
   3 | 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
   4 | 1, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
   5 | 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
   6 | 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
   7 | 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
   8 | 1, 3, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
   9 | 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0
  10 | 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0
  11 | 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0
  12 | 1, 2, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0
  13 | 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0
  14 | 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0
  15 | 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0
  16 | 1, 4, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0
  17 | 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0
  18 | 1, 1, 2, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1
  ---------------------------------------------------------
A(18,2) = 1, because 2^1 divides 18, but 2^2 does not. A(18,3) = 2, because 3^2 divides 18 (but 3^3 does not). A(18,4) = 0, because 4^0 (= 1) divides 18, but 4^1 does not. A(18,18) = 1, because 18^1 divides 18, but 18^2 does not.
A(2,18) = 0, because 18^0 divides 2, but 18^1 does not.

Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 antidiagonals of array

Cf. A286562 (transpose), A286563 (lower triangular region), A286564 (lower triangular region reversed).
Cf. A169594 (row sums), also A168512, A178638, A186643.
Cf. also array A286156.

Table[Function[m, If[k == 1, 1, IntegerExponent[m, k]]][n - k + 1], {n, 15}, {k, n}] // TableForm (* Michael De Vlieger, May 20 2017 *)

A286561(n,k) = if(1==k, 1, valuation(n, k)); \\ Antti Karttunen, May 27 2017

def a(n, k):
    i=1
    if k==1: return 1
    while n%(k**i)==0:
        i+=1
    return i-1
for n in range(1, 21): print([a(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, May 20 2017

(define (A286561 n) (A286561bi (A002260 n) (A004736 n)))
(define (A286561bi row col) (if (= 1 col) 1 (let loop ((i 1)) (if (not (zero? (modulo row (expt col i)))) (- i 1) (loop (+ 1 i))))))

A130517 Triangle read by rows: row n counts down from n in steps of 2, then counts up the remaining elements in the set {1,2,...,n}, again in steps of 2.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 4, 2, 1, 3, 5, 3, 1, 2, 4, 6, 4, 2, 1, 3, 5, 7, 5, 3, 1, 2, 4, 6, 8, 6, 4, 2, 1, 3, 5, 7, 9, 7, 5, 3, 1, 2, 4, 6, 8, 10, 8, 6, 4, 2, 1, 3, 5, 7, 9, 11, 9, 7, 5, 3, 1, 2, 4, 6, 8, 10, 12, 10, 8, 6, 4, 2, 1, 3, 5, 7, 9, 11, 13, 11, 9, 7, 5, 3, 1, 2, 4, 6, 8, 10, 12, 14, 12, 10

Offset: 1

Omar E. Pol, Aug 08 2007

Triangle read by rows in which row n lists the number of pairs of states of the subshells of the n-th shell of the nuclear shell model ordered by energy level in increasing order.
Row n lists a permutation of the first n positive integers.
If n is odd then row n lists the first (n+1)/2 odd numbers in decreasing order together with the first (n-1)/2 positive even numbers.
If n is even then row n lists the first n/2 even numbers in decreasing order together with the first n/2 odd numbers.
Row n >= 2, with its floor(n/2) last numbers taken as negative, lists the n different eigenvalues (in decreasing order) of the odd graph O(n). The odd graph O(n) has the (n-1)-subsets of a (2*n-1)-set as vertices, with two (n-1)-subsets adjacent if and only if they are disjoint. For example, O(3) is isomorphic to the Petersen graph. - Miquel A. Fiol, Apr 07 2024

			A geometric model of the atomic nucleus:
......-------------------------------------------------
......|...-----------------------------------------...|
......|...|...---------------------------------...|...|
......|...|...|...-------------------------...|...|...|
......|...|...|...|...-----------------...|...|...|...|
......|...|...|...|...|...---------...|...|...|...|...|
......|...|...|...|...|...|...-...|...|...|...|...|...|
......i...h...g...f...d...p...s...p...d...f...g...h...i
......|...|...|...|...|...|.......|...|...|...|...|...|
......|...|...|...|...|.......1.......|...|...|...|...|
......|...|...|...|.......2.......1.......|...|...|...|
......|...|...|.......3.......1.......2.......|...|...|
......|...|.......4.......2.......1.......3.......|...|
......|.......5.......3.......1.......2.......4.......|
..........6.......4.......2.......1.......3.......5....
......7.......5.......3.......1.......2.......4.......6
.......................................................
...13/2.11/2.9/2.7/2.5/2.3/2.1/2.1/2.3/2.5/2.7/2.9/2.11/2
......|...|...|...|...|...|...|...|...|...|...|...|...|
......|...|...|...|...|...|...-----...|...|...|...|...|
......|...|...|...|...|...-------------...|...|...|...|
......|...|...|...|...---------------------...|...|...|
......|...|...|...-----------------------------...|...|
......|...|...-------------------------------------...|
......|...---------------------------------------------
.
Triangle begins:
   1;
   2, 1;
   3, 1, 2;
   4, 2, 1, 3;
   5, 3, 1, 2, 4;
   6, 4, 2, 1, 3, 5;
   7, 5, 3, 1, 2, 4, 6;
   8, 6, 4, 2, 1, 3, 5, 7;
   9, 7, 5, 3, 1, 2, 4, 6, 8;
  10, 8, 6, 4, 2, 1, 3, 5, 7, 9;
  ...
Also:
                     1;
                   2,  1;
                 3,  1,  2;
               4,  2,  1,  3;
             5,  3,  1,  2,  4;
           6,  4,  2,  1,  3,  5;
         7,  5,  3,  1,  2,  4,  6;
       8,  6,  4,  2,  1,  3,  5,  7;
     9,  7,  5,  3,  1,  2,  4,  6,  8;
  10,  8,  6,  4,  2,  1,  3,  5,  7,  9;
  ...
In this view each column contains the same numbers.
From _Miquel A. Fiol_, Apr 07 2024: (Start)
Eigenvalues of the odd graphs O(n) for n=2..10:
   2, -1;
   3,  1, -2;
   4,  2, -1, -3;
   5,  3,  1, -2, -4;
   6,  4,  2, -1, -3, -5;
   7,  5,  3,  1, -2, -4, -6;
   8,  6,  4,  2, -1, -3, -5, -7;
   9,  7,  5,  3,  1, -2, -4, -6, -8;
  10,  8,  6,  4,  2, -1, -3, -5, -7, -9;
... (End)

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
N. Bigss, Algebraic Graph Theory, Cambridge Univ. Press, Cambridge, 1974.
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions, 2012, arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Odd graph

Absolute values of A056951. Column 1 is A000027. Row sums are in A000217.
Cf. A130556, A130598, A130602.
Other versions are A004736, A212121, A213361, A213371.
Cf. A028310 (right edge), A000012 (central terms), A220073 (mirrored), A220053 (partial sums in rows), A375303.

a130517 n k = a130517_tabl !! (n-1) !! (k-1)
a130517_row n = a130517_tabl !! (n-1)
a130517_tabl = iterate (\row -> (head row + 1) : reverse row) [1]
-- Reinhard Zumkeller, Dec 03 2012

A130517 := proc(n,k)
     if k <= (n+1)/2 then
        n-2*(k-1) ;
    else
        1-n+2*(k-1) ;
    end if;
end proc: # R. J. Mathar, Jul 21 2012

t[n_, 1] := n; t[n_, n_] := n-1; t[n_, k_] := Abs[2*k-n - If[2*k <= n+1, 2, 1]]; Table[t[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 03 2013, from abs(A056951) *)

a130517_row(n) = my(v=vector(n), s=1, n1=0, n2=n+1); forstep(k=n, 1,-1, s=-s; if(s>0, n2--; v[n2]=k, n1++; v[n1]=k)); v \\ Hugo Pfoertner, Aug 26 2024

a(n) = A162630(n)/2. - Omar E. Pol, Sep 02 2012
T(1,1) = 1; for n > 1: T(n,1) = T(n-1,1)+1 and T(n,k) = T(n-1,n-k+1), 1 < k <= n. - Reinhard Zumkeller, Dec 03 2012
From Boris Putievskiy, Jan 16 2013: (Start)
a(n) = |2*A000027(n) - A003056(n)^2 - 2*A003056(n) - 3| + floor((2*A000027(n) - A003056(n)^2 - A003056(n))/(A003056(n)+3)).
a(n) = |2*n - t^2 - 2*t - 3| + floor((2*n - t^2 - t)/(t+3)) where t = floor((-1+sqrt(8*n-7))/2). (End)

A059896 The set of Fermi-Dirac factors of A(n,k) is the union of the Fermi-Dirac factors of n and k. Symmetric square array read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 6, 6, 4, 5, 8, 3, 8, 5, 6, 10, 12, 12, 10, 6, 7, 6, 15, 4, 15, 6, 7, 8, 14, 6, 20, 20, 6, 14, 8, 9, 8, 21, 24, 5, 24, 21, 8, 9, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 11, 10, 27, 8, 35, 6, 35, 8, 27, 10, 11, 12, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 12, 13, 24

Offset: 1

Marc LeBrun, Feb 06 2001

Every positive integer, m, is the product of a unique subset, S(m), of the numbers listed in A050376 (primes, squares of primes etc.) The Fermi-Dirac factors of m are the members of S(m). So T(n,k) is the product of the members of (S(n) U S(k)).
Old name: Table a(i,j) = product prime(k)^(Ei(k) OR Ej(k)) where Ei and Ej are the vectors of exponents in the prime factorizations of i and j; OR is the bitwise operation on binary representation of the exponents.
Analogous to LCM, with OR replacing MAX.
A003418-analog seems to be A066616. - Antti Karttunen, Apr 12 2017
Considered as a binary operation, the result is the lowest common multiple of the squarefree parts of its operands multiplied by the square of the operation's result when applied to the square roots of the square parts of its operands. - Peter Munn, Mar 02 2022

			A(864,1944) = A(2^5*3^3,2^3*3^5) = 2^(5 OR 3) * 3^(3 OR 5) = 2^7*3^7 = 279936.
The top left 12 X 12 corner of the array:
   1,  2,  3,  4,  5,  6,  7,  8,   9,  10,  11,  12
   2,  2,  6,  8, 10,  6, 14,  8,  18,  10,  22,  24
   3,  6,  3, 12, 15,  6, 21, 24,  27,  30,  33,  12
   4,  8, 12,  4, 20, 24, 28,  8,  36,  40,  44,  12
   5, 10, 15, 20,  5, 30, 35, 40,  45,  10,  55,  60
   6,  6,  6, 24, 30,  6, 42, 24,  54,  30,  66,  24
   7, 14, 21, 28, 35, 42,  7, 56,  63,  70,  77,  84
   8,  8, 24,  8, 40, 24, 56,  8,  72,  40,  88,  24
   9, 18, 27, 36, 45, 54, 63, 72,   9,  90,  99, 108
  10, 10, 30, 40, 10, 30, 70, 40,  90,  10, 110, 120
  11, 22, 33, 44, 55, 66, 77, 88,  99, 110,  11, 132
  12, 24, 12, 12, 60, 24, 84, 24, 108, 120, 132,  12

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of the array
Eric Weisstein's World of Mathematics, Square Part.
Eric Weisstein's World of Mathematics, Squarefree Part.

Cf. A003418, A003990, A007947, A028233, A028234, A066616, A284576.
Sequences used in a definition of this sequence: A003986, A000188/A007913/A008833, A052330/A052331.
Has simple/very significant relationships with A003961, A059895/A059897, A225546, A267116.

a[i_, i_] := i;
a[i_, j_] := Module[{f1 = FactorInteger[i], f2 = FactorInteger[j], e1, e2}, e1[] = 0; Scan[(e1[#[[1]]] = #[[2]])&, f1]; e2[] = 0; Scan[(e2[#[[1]]] = #[[2]])&, f2]; Times @@ (#^BitOr[e1[#], e2[#]]& /@ Union[f1[[All, 1]], f2[[All, 1]]])];
Table[a[i - j + 1, j], {i, 1, 15}, {j, 1, i}] // Flatten (* Jean-François Alcover, Jun 19 2018 *)

A059896(n,k) = if(n==k, n, lcm(core(n),core(k)) * A059896(core(n,1)[2],core(k,1)[2])^2) \\ Peter Munn, Mar 07 2022

(define (A059896 n) (A059896bi (A002260 n) (A004736 n)))
(define (A059896bi a b) (let loop ((a a) (b b) (m 1)) (cond ((= 1 a) (* m b)) ((= 1 b) (* m a)) ((equal? (A020639 a) (A020639 b)) (loop (A028234 a) (A028234 b) (* m (expt (A020639 a) (A003986bi (A067029 a) (A067029 b)))))) ((< (A020639 a) (A020639 b)) (loop (/ a (A028233 a)) b (* m (A028233 a)))) (else (loop a (/ b (A028233 b)) (* m (A028233 b)))))))
;; Antti Karttunen, Apr 11 2017

From Antti Karttunen, Apr 11 2017: (Start)
A(x,y) = A059895(x,y) * A059897(x,y).
A(x,y) * A059895(x,y) = x*y.
(End).
From Peter Munn, Mar 02 2022: (Start)
OR denotes the bitwise operation (A003986).
Limited multiplicative property: if gcd(n_1*k_1, n_2*k_2) = 1 then A(n_1*n_2, k_1*k_2) = A(n_1, k_1) * A(n_2, k_2).
For prime p, A(p^e_1, p^e_2) = p^(e_1 OR e_2).
A(n, A(m, k)) = A(A(n, m), k).
A(n, k) = A(k, n).
A(n, 1) = A(n, n) = n.
A(n^2, k^2) = A(n, k)^2.
A(n, k) = A(A007913(n), A007913(k)) * A(A008833(n), A008833(k)) = lcm(A007913(n), A007913(k)) * A(A000188(n), A000188(k))^2.
A007947(A(n, k)) = A007947(n*k).
Isomorphism: A(A052330(n), A052330(k)) = A052330(n OR k).
Equivalently, A(n, k) = A052330(A052331(n) OR A052331(k)).
A(A003961(n), A003961(k)) = A003961(A(n, k)).
A(A225546(n), A225546(k)) = A225546(A(n, k)).
(End)

New name from Peter Munn, Mar 02 2022

A193842 Triangular array: the fission of the polynomial sequence ((x+1)^n: n >= 0) by the polynomial sequence ((x+2)^n: n >= 0). (Fission is defined at Comments.)

Original entry on oeis.org

1, 1, 4, 1, 7, 13, 1, 10, 34, 40, 1, 13, 64, 142, 121, 1, 16, 103, 334, 547, 364, 1, 19, 151, 643, 1549, 2005, 1093, 1, 22, 208, 1096, 3478, 6652, 7108, 3280, 1, 25, 274, 1720, 6766, 17086, 27064, 24604, 9841, 1, 28, 349, 2542, 11926, 37384, 78322, 105796

Offset: 0

Clark Kimberling, Aug 07 2011

Suppose that p = p(n)*x^n + p(n-1)*x^(n-1) + ... + p(1)*x + p(0) is a polynomial and that Q is a sequence of polynomials:
...
q(k,x) = t(k,0)*x^k + t(k,1)*x^(k-1) + ... + t(k,k-1)*x + t(k,k),
...
for k = 0, 1, 2, ...  The Q-downstep of p is the polynomial given by
...
D(p) = p(n)*q(n-1,x) + p(n-1)*q(n-2,x) + ... + p(1)*q(0,x). (Note that p(0) does not appear. "Q-downstep" as just defined differs slightly from "Q-downstep" as defined for a different purpose at A193649.)
...
Now suppose that P = (p(n,x): n >= 0) and Q = (q(n,x): n >= 0) are sequences of polynomials, where n indicates degree.  The fission of P by Q, denoted by P^^Q, is introduced here as the sequence W = (w(n,x): n >= 0) of polynomials defined by w(0,x) = 1 and w(n,x) = D(p(n+1,x)).
...
Strictly speaking, ^^ is an operation on sequences of polynomials.  However, if P and Q are regarded as numerical triangles (of coefficients of polynomials), then ^^ can be regarded as an operation on numerical triangles.  In this case, row n of P^^Q, for n > 0, is given by the matrix product P(n+1)*QQ(n), where P(n+1) =(p(n+1,n+1), p(n+1,n), ..., p(n+1,2), p(n+1,1)) and QQ(n) is the (n+1)-by-(n+1) matrix given by
...
q(n,0) .. q(n,1)............. q(n,n-1) .... q(n,n)
0 ....... q(n-1,0)........... q(n-1,n-2)... q(n-1,n-1)
0 ....... 0.................. q(n-2,n-3) .. q(n-2,n-2)
...
0 ....... 0.................. q(1,0) ...... q(1,1)
0 ....... 0 ................. 0 ........... q(0,0).
Here, the polynomial q(k,x) is taken to be
q(k,0)*x^k + q(k,1)x^(k-1) + ... + q(k,k)*x + q(k,k);
i.e., "q" is used instead of "t".
...
Example:  Let p(n,x) = (x+1)^n and q(n,x) = (x+2)^n.  Then
...
w(0,x) = 1 by the definition of W,
w(1,x) = D(p(2,x)) = 1*(x+2) + 2*1 = x + 4,
w(2,x) = D(p(3,x)) = 1*(x^2+4*x+4) + 3*(x+2) + 3*1 = x^2 + 7*x + 13,
w(3,x) = D(p(4,x)) = 1*(x^3+6*x^2+12*x+8) + 4*(x^2+4x+4) + 6*(x+2) + 4*1 = x^3 + 10*x^2 + 34*x + 40.
...
From these first 4 polynomials in the sequence P^^Q, we can write the first 4 rows of P^^Q when P, Q, and P^^Q are regarded as triangles:
1
1...4
1...7....13
1...10...34...40
...
In the following examples, r(P^^Q) is the mirror of P^^Q, obtained by reversing the rows of P^^Q.  Let u denote the polynomial x^n + x^(n-1) + ... + x + 1.
...
..P........Q...........P^^Q........r(P^^Q)
(x+1)^n....(x+2)^n.....A193842.....A193843
(x+1)^n....(x+1)^n.....A193844.....A193845
(x+2)^n....(x+1)^n.....A193846.....A193847
(2x+1)^n...(x+1)^n.....A193856.....A193857
(x+1)^n....(2x+1)^n....A193858.....A193859
(x+1)^n.......u........A054143.....A104709
..u........(x+1)^n.....A074909.....A074909
..u...........u........A002260.....A004736
(x+2)^n.......u........A193850.....A193851
..u.........(x+2)^n....A193844.....A193845
(2x+1)^n......u........A193860.....A193861
..u.........(2x+1)^n...A115068.....A193862
...
Regarding A193842,
col 1 ...... A000012
col 2 ...... A016777
col 3 ...... A081271
w(n,n) ..... A003462
w(n,n-1) ... A014915

			First six rows, for 0 <= k <= n and 0 <= n <= 5:
  1
  1...4
  1...7....13
  1...10...34....40
  1...13...64....142...121
  1...16...103...334...547...364

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Digital Library of Mathematical Functions, Hypergeometric function, analytic properties.
Clark Kimberling, Fusion, Fission, and Factors, Fib. Q., 52(3) (2014), 195-202.

Cf. A193722 (fusion of P by Q), A193649 (Q-residue), A193843 (mirror of A193842).

[ (&+[3^(k-j)*Binomial(n-j,k-j): j in [0..k]]): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 18 2020

fission := proc(p, q, n) local d, k;
p(n+1,0)*q(n,x)+add(coeff(p(n+1,x),x^k)*q(n-k,x), k=1..n);
seq(coeff(%,x,n-k), k=0..n) end:
A193842_row := n -> fission((n,x) -> (x+1)^n, (n,x) -> (x+2)^n, n);
for n from 0 to 5 do A193842_row(n) od; # Peter Luschny, Jul 23 2014
# Alternatively:
p := (n,x) -> add(x^k*(1+3*x)^(n-k),k=0..n): for n from 0 to 7 do [n], PolynomialTools:-CoefficientList(p(n,x), x) od; # Peter Luschny, Jun 18 2017

(* First program *)
z = 10;
p[n_, x_] := (x + 1)^n;
q[n_, x_] := (x + 2)^n
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193842 *)
TableForm[Table[h[n], {n, 0, z}]]  (* A193843 *)
Flatten[Table[h[n], {n, -1, z}]]
(* Second program *)
Table[SeriesCoefficient[((x+3)^(n+1) -1)/(x+2), {x,0,n-k}], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 18 2020 *)

T(n,k) = sum(j=0,k, 3^(k-j)*binomial(n-j,k-j)); \\ G. C. Greubel, Feb 18 2020

from mpmath import mp, hyp2f1
mp.dps = 100; mp.pretty = True
def T(n,k):
    return 3^k*binomial(n,k)*hyp2f1(1,-k,-n,1/3)-0^(n-k)//2
for n in range(7):
    print([int(T(n,k)) for k in (0..n)]) # Peter Luschny, Jul 23 2014

# Second program using the 'fission' operation.
def fission(p, q, n):
    F = p(n+1,0)*q(n,x)+add(expand(p(n+1,x)).coefficient(x,k)*q(n-k,x) for k in (1..n))
    return [expand(F).coefficient(x,n-k) for k in (0..n)]
A193842_row = lambda k: fission(lambda n,x: (x+1)^n, lambda n,x: (x+2)^n, k)
for n in range(7): A193842_row(n) # Peter Luschny, Jul 23 2014

From Peter Bala, Jul 16 2013: (Start)
T(n,k) = Sum_{i = 0..k} 3^(k-i)*binomial(n-i,k-i).
O.g.f.: 1/((1 - x*t)*(1 - (1 + 3*x)*t)) = 1 + (1 + 4*x)*t + (1 + 7*x + 13*x^2)*t^2 + ....
The n-th row polynomial is R(n,x) = (1/(2*x + 1))*((3*x + 1)^(n+1) - x^(n+1)). (End)
T(n,k) = T(n-1,k) + 4*T(n-1,k-1) - T(n-2,k-1) - 3*T(n-2,k-2), T(0,0) = 1, T(1,0) = 1, T(1,1) = 4, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Jan 17 2014
T(n,k) = 3^k * C(n,k) * hyp2F1(1, -k, -n, 1/3) with or without the additional term -0^(n-k)/2 depending on the exact definition of the hypergeometric function used. Compare formulas 15.2.5 and 15.2.6 in the DLMF reference. - Peter Luschny, Jul 23 2014

Name and Comments edited by Petros Hadjicostas, Jun 05 2020

A051162 Triangle T(n,k) = n+k, n >= 0, 0 <= k <= n.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 3, 4, 5, 6, 4, 5, 6, 7, 8, 5, 6, 7, 8, 9, 10, 6, 7, 8, 9, 10, 11, 12, 7, 8, 9, 10, 11, 12, 13, 14, 8, 9, 10, 11, 12, 13, 14, 15, 16, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

Offset: 0

N. J. A. Sloane

Row sums are A045943 = triangular matchstick numbers: 3n(n+1)/2. This was independently noted by me and, without cross-reference, as a comment on A045943, by Jon Perry, Jan 15 2004. - Jonathan Vos Post, Nov 09 2007
In partitions of n into distinct parts having maximal size, a(n) is the greatest number, see A000009. - Reinhard Zumkeller, Jun 13 2009
Row sums of reciprocals of terms in this triangle converge to log(2). See link to Eric Naslund's answer. - Mats Granvik, Mar 07 2013
T(n,k) satisfies the cubic equation T(n,k)^3 + 3*A025581(n, k)*T(n,k) - 4*A105125(n,k) = 0. This is a problem similar to the one posed by François Viète (Vieta) mentioned in a comment on A025581. Here the problem is to determine for a rectangle (a, b), with a > b >= 1, from the given values for a^3 + b^3 and a - b the value of a + b. Here for nonnegative integers a = n and b = k. - Wolfdieter Lang, May 15 2015
If we subtract 1 from every term the result is essentially A213183. - N. J. A. Sloane, Apr 28 2020

			The triangle  T(n, k) starts:
n\k  0  1  2  3  4  5  6  7  8  9 10 ...
0:   0
1:   1  2
2:   2  3  4
3:   3  4  5  6
4:   4  5  6  7  8
5:   5  6  7  8  9 10
6:   6  7  8  9 10 11 12
7:   7  8  9 10 11 12 13 14
8:   8  9 10 11 12 13 14 15 16
9:   9 10 11 12 13 14 15 16 17 18
10: 10 11 12 13 14 15 16 17 18 19 20
... reformatted. - _Wolfdieter Lang_, May 15 2015

Reinhard Zumkeller, Rows n=0..100 of triangle, flattened
Eric Naslund, Euler-Mascheroni constant expression, further simplification
Dmitry A. Zaitsev, A generalized neighborhood for cellular automata, Theoretical Computer Science, 2016, Volume 666, 1 March 2017, Pages 21-35.

Cf. A000009, A000217, A002260, A025581, A004736, A045943, A105125, A213183.
Cf. also A008585 (central terms), A005843 (right edge).
Cf. also A002262, A001477, A003056.

a051162 n k = a051162_tabl !! n !! k
a051162_row n = a051162_tabl !! n
a051162_tabl = iterate (\xs@(x:_) -> (x + 1) : map (+ 2) xs) [0]
-- Reinhard Zumkeller, Sep 17 2014, Oct 02 2012, Apr 23 2012

seq(seq(r+c, c=0..r),r=0..10); # Robert Israel, May 21 2015

With[{c=Range[0,20]}, Flatten[Table[Take[c,{n,2n-1}], {n,11}]]] (* Harvey P. Dale, Nov 19 2011 *)

for(n=0,10,for(k=0,n,print1(n+k,", "))) \\ Derek Orr, May 19 2015

T(n, k) = n + k, 0 <= k <= n.
a(n-1) = 2*A002260(n) + A004736(n) - 3, n > 0. - Boris Putievskiy, Mar 12 2012
a(n-1) = (t - t^2+ 2n-2)/2, where t = floor((-1+sqrt(8*n-7))/2), n > 0. - Robert G. Wilson v and Boris Putievskiy, Mar 14 2012
From Robert Israel, May 21 2015: (Start)
a(n) = A003056(n) + A002262(n).
G.f.: x/(1-x)^2 + (1-x)^(-1)*Sum(j>=1, (1-j)*x^A000217(j)). The sum is related to Jacobi Theta functions. (End)
G.f. as triangle: (x + (2 - 3*x)*x*y)/((1 - x)^2*(1 - x*y)^2). - Stefano Spezia, Apr 22 2024

A193091 Augmentation of the triangular array A158405. See Comments.

Original entry on oeis.org

1, 1, 3, 1, 6, 14, 1, 9, 37, 79, 1, 12, 69, 242, 494, 1, 15, 110, 516, 1658, 3294, 1, 18, 160, 928, 3870, 11764, 22952, 1, 21, 219, 1505, 7589, 29307, 85741, 165127, 1, 24, 287, 2274, 13355, 61332, 224357, 638250, 1217270, 1, 27, 364, 3262, 21789, 115003

Offset: 0

Clark Kimberling, Jul 30 2011

Suppose that P is an infinite triangular array of numbers:
p(0,0)
p(1,0)...p(1,1)
p(2,0)...p(2,1)...p(2,2)
p(3,0)...p(3,1)...p(3,2)...p(3,3)...
...
Let w(0,0)=1, w(1,0)=p(1,0), w(1,1)=p(1,1), and define
    W(n)=(w(n,0), w(n,1), w(n,2),...w(n,n-1), w(n,n)) recursively by W(n)=W(n-1)*PP(n), where PP(n) is the n X (n+1) matrix given by
...
row 0 ... p(n,0) ... p(n,1) ...... p(n,n-1) ... p(n,n)
row 1 ... 0 ..... p(n-1,0) ..... p(n-1,n-2) .. p(n-1,n-1)
row 2 ... 0 ..... 0 ............ p(n-2,n-3) .. p(n-2,n-2)
...
row n-1 . 0 ..... 0 ............. p(2,1) ..... p(2,2)
row n ... 0 ..... 0 ............. p(1,0) ..... p(1,1)
...
The augmentation of P is here introduced as the triangular array whose n-th row is W(n), for n>=0. The array P may be represented as a sequence of polynomials; viz., row n is then the vector of coefficients:  p(n,0), p(n,1),...,p(n,n), from p(n,0)*x^n+p(n,1)*x^(n-1)+...+p(n,n). For example, (C(n,k)) is represented by ((x+1)^n); using this choice of P (that is, Pascal's triangle), the augmentation of P is calculated one row at a time, either by the above matrix products or by polynomial substitutions in the following manner:
...
row 0 of W:  1, by decree
row 1 of W:  1 augments to 1,1
...polynomial version:  1 -> x+1
row 2 of W:  1,1 augments to 1,3,2
...polynomial version:  x+1 -> (x^2+2x+1)+(x+1)=x^2+3x+2
row 3 to W:  1,3,2 augments to 1,6,11,6
...polynomial version:
    x^2+3x+2 -> (x+1)^3+3(x+1)^2+2(x+1)=(x+1)(x+2)(x+3)
...
Examples of augmented triangular arrays:
(p(n,k)=1) augments to A009766, Catalan triangle.
Catalan triangle augments to A193560.
Pascal triangle augments to A094638, Stirling triangle.
A002260=((k+1)) augments to A023531.
A154325 augments to A033878.
A158405 augments to A193091.
((k!)) augments to A193092.
A094727 augments to A193093.
A130296 augments to A193094.
A004736 augments to A193561.
...
Regarding the specific augmentation W=A193091: w(n,n)=A003169.
From Peter Bala, Aug 02 2012: (Start)
This is the table of g(n,k) in the notation of Carlitz (p. 124). The triangle enumerates two-line arrays of positive integers
............a_1 a_2 ... a_n..........
............b_1 b_2 ... b_n..........
such that
1) max(a_i, b_i) <= min(a_(i+1), b_(i+1)) for 1 <= i <= n-1
2) max(a_i, b_i) <= i for 1 <= i <= n
3) max(a_n, b_n) = k.
See A071948 and A211788 for other two-line array enumerations.
(End)

			The triangle P, at A158405, is given by rows
  1
  1...3
  1...3...5
  1...3...5...7
  1...3...5...7...9...
The augmentation of P is the array W starts with w(0,0)=1, by definition of W.
Successive polynomials (rows of W) arise from P as shown here:
  ...
  1->x+3, so that W has (row 1)=(1,3);
  ...
  x+3->(x^2+3x+5)+3*(x+3), so that W has (row 2)=(1,6,14);
  ...
  x^2+6x+14->(x^3+3x^2+5x+7)+6(x^2+3x+5)+14(x+3), so that (row 3)=(1,9,37,79).
  ...
First 7 rows of W:
  1
  1    3
  1    6    14
  1    9    37    79
  1   12    69   242    494
  1   15   110   516   1658    3294
  1   18   160   928   3870   11764   22952

L. Carlitz, Enumeration of two-line arrays, Fib. Quart., Vol. 11 Number 2 (1973), 113-130.

Cf. A003169, A193092, A193093, A193094.

Cf. A071948, A211788.

p[n_, k_] := 2 k + 1
Table[p[n, k], {n, 0, 5}, {k, 0, n}] (* A158405 *)
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 6}]] (* A193091 *)
Flatten[Table[v[n], {n, 0, 9}]]

From Peter Bala, Aug 02 2012: (Start)
T(n,k) = (n-k+1)/n*Sum_{i=0..k} C(n+1,n-k+i+1)*C(2*n+i+1,i) for 0 <= k <= n.
Recurrence equation: T(n,k) = Sum_{i=0..k} (2*k-2*i+1)*T(n-1,i).
(End)

A049581 Table T(n,k) = |n-k| read by antidiagonals (n >= 0, k >= 0).

Original entry on oeis.org

0, 1, 1, 2, 0, 2, 3, 1, 1, 3, 4, 2, 0, 2, 4, 5, 3, 1, 1, 3, 5, 6, 4, 2, 0, 2, 4, 6, 7, 5, 3, 1, 1, 3, 5, 7, 8, 6, 4, 2, 0, 2, 4, 6, 8, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 10, 8, 6, 4, 2, 0, 2, 4, 6, 8, 10, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 12, 10, 8, 6, 4, 2, 0, 2, 4, 6, 8, 10, 12

Offset: 0

N. J. A. Sloane

Commutative non-associative operator with identity 0. T(nx,kx) = x T(n,k). A multiplicative analog is A089913. - Marc LeBrun, Nov 14 2003
For the characteristic polynomial of the n X n matrix M_n with entries M_n(i, j) = |i-j| see A203993. - Wolfdieter Lang, Feb 04 2018
For the determinant of the n X n matrix M_n with entries M_n(i, j) = |i-j| see A085750. - Bernard Schott, May 13 2020
a(n) = 0 iff n = 4 times triangular number (A046092). - Bernard Schott, May 13 2020

			Displayed as a triangle t(n, k):
  n\k   0 1 2 3 4 5 6 7 8 9 10 ...
  0:    0
  1:    1 1
  2:    2 0 2
  3:    3 1 1 3
  4:    4 2 0 2 4
  5:    5 3 1 1 3 5
  6:    6 4 2 0 2 4 6
  7:    7 5 3 1 1 3 5 7
  8:    8 6 4 2 0 2 4 6 8
  9:    9 7 5 3 1 1 3 5 7 9
  10:  10 8 6 4 2 0 2 4 6 8 10
... reformatted by _Wolfdieter Lang_, Feb 04 2018
Displayed as a table:
  0 1 2 3 4 5 6 ...
  1 0 1 2 3 4 5 ...
  2 1 0 1 2 3 4 ...
  3 2 1 0 1 2 3 ...
  4 3 2 1 0 1 2 ...
  5 4 3 2 1 0 1 ...
  6 5 4 3 2 1 0 ...
  ...

Peter Kagey, Rows n = 0..125 of triangle, flattened

Cf. A003989, A003990, A003991, A003056, A004247, A002260, A004736.

Cf. A089913. Apart from signs, same as A114327. A203993.

a := Flat(List([0..12],n->List([0..n],k->Maximum(k,n-k)-Minimum(k,n-k)))); # Muniru A Asiru, Jan 26 2018

[[Abs(n-2*k): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Jun 07 2019

seq(seq(abs(n-2*k),k=0..n),n=0..12); # Robert Israel, Sep 30 2015

Table[Abs[(n-k) -k], {n,0,12}, {k,0,n}]//Flatten (* Michael De Vlieger, Sep 29 2015 *)
Table[Join[Range[n,0,-2],Range[If[EvenQ[n],2,1],n,2]],{n,0,12}]//Flatten (* Harvey P. Dale, Sep 18 2023 *)

a(n) = abs(2*(n+1)-binomial((sqrtint(8*(n+1))+1)\2, 2)-(binomial(1+floor(1/2 + sqrt(2*(n+1))), 2))-1);
vector(100, n , a(n-1)) \\ Altug Alkan, Sep 29 2015

{t(n,k) = abs(n-2*k)}; \\ G. C. Greubel, Jun 07 2019

from math import isqrt
def A049581(n): return abs((k:=n+1<<1)-((m:=isqrt(k))+(k>m*(m+1)))**2-1) # Chai Wah Wu, Nov 09 2024

[[abs(n-2*k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jun 07 2019

G.f.: (x + y - 4*x*y + x^2*y + x*y^2)/((1-x)^2*(1-y)^2*(1-x*y)) = (x/(1-x)^2 + y/(1-y)^2)/(1-x*y). T(n,0) = T(0,n) = n; T(n+1,k+1) = T(n,k). - Franklin T. Adams-Watters, Feb 06 2006
a(n) = |A002260(n+1)-A004736(n+1)| or a(n) = |((n+1)-t*(t+1)/2) - ((t*t+3*t+4)/2-(n+1))| where t = floor((-1+sqrt(8*(n+1)-7))/2). - Boris Putievskiy, Dec 24 2012; corrected by Altug Alkan, Sep 30 2015
From Robert Israel, Sep 30 2015: (Start)
If b(n) = a(n+1) - 2*a(n) + a(n-1), then for n >= 3 we have
  b(n) = -1 if n = (j^2+5j+4)/2 for some integer j >= 1
  b(n) = -3 if n = (j^2+5j+6)/2 for some integer j >= 0
  b(n) = 4 if n = 2j^2 + 6j + 4 for some integer j >= 0
  b(n) = 2 if n = 2j^2 + 8j + 7 or 2j^2 + 8j + 8 for some integer j >= 0
  b(n) = 0 otherwise. (End)
Triangle t(n,k) = max(k, n-k) - min(k, n-k). - Peter Luschny, Jan 26 2018
Triangle t(n, k) = |n - 2*k| for n >= 0, k = 0..n. See the Maple and Mathematica programs. Hence t(n, k)= t(n, n-k). - Wolfdieter Lang, Feb 04 2018
a(n) = |t^2 - 2*n - 1|, where t = floor(sqrt(2*n+1) + 1/2). - Ridouane Oudra, Jun 07 2019; Dec 11 2020
As a rectangle, T(n,k) = |n-k| = max(n,k) - min(n,k). - Clark Kimberling, May 11 2020

A059379 Array of values of Jordan function J_k(n) read by antidiagonals (version 1).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 2, 8, 7, 1, 4, 12, 26, 15, 1, 2, 24, 56, 80, 31, 1, 6, 24, 124, 240, 242, 63, 1, 4, 48, 182, 624, 992, 728, 127, 1, 6, 48, 342, 1200, 3124, 4032, 2186, 255, 1, 4, 72, 448, 2400, 7502, 15624, 16256, 6560, 511, 1, 10, 72, 702, 3840

Offset: 1

N. J. A. Sloane, Jan 28 2001

			Array begins:
  1,  1,  2,   2,   4,    2,    6,    4,   6,  4, 10, 4, ...
  1,  3,  8,  12,  24,   24,   48,   48,  72, 72, ...
  1,  7, 26,  56, 124,  182,  342,  448, 702, ...
  1, 15, 80, 240, 624, 1200, 2400, 3840, ...

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
R. Sivaramakrishnan, "The many facets of Euler's totient. II. Generalizations and analogues", Nieuw Arch. Wisk. (4) 8 (1990), no. 2, 169-187.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000

See A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5). Columns give A000225, A024023, A020522, A024049, A059387, etc.

Main diagonal gives A067858.

J := proc(n,k) local i,p,t1,t2; t1 := n^k; for p from 1 to n do if isprime(p) and n mod p = 0 then t1 := t1*(1-p^(-k)); fi; od; t1; end;
#alternative
A059379 := proc(n,k)
    add(d^k*numtheory[mobius](n/d),d=numtheory[divisors](n)) ;
end proc:
seq(seq(A059379(d-k,k),k=1..d-1),d=2..12) ; # R. J. Mathar, Nov 23 2018

JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/#]&]/;(n>0)&&IntegerQ[n];
A004736[n_]:=Binomial[Floor[3/2+Sqrt[2*n]],2]-n+1;
A002260[n_]:=n-Binomial[Floor[1/2+Sqrt[2*n]],2];
A059379[n_]:=JordanTotient[A004736[n],A002260[n]]; (* Enrique Pérez Herrero, Dec 19 2010 *)

jordantot(n,k)=sumdiv(n,d,d^k*moebius(n/d));
A002260(n)=n-binomial(floor(1/2+sqrt(2*n)),2);
A004736(n)=binomial(floor(3/2+sqrt(2*n)),2)-n+1;
A059379(n)=jordantot(A004736(n),A002260(n)); \\ Enrique Pérez Herrero, Jan 08 2011

from functools import cache
def MoebiusTrans(a, i):
    @cache
    def mb(n, d = 1):
          return d % n and -mb(d, n % d < 1) + mb(n, d + 1) or 1 // n
    def mob(m, n): return mb(m // n) if m % n == 0 else 0
    return sum(mob(i, d) * a(d) for d in range(1, i + 1))
def Jrow(n, size):
    return [MoebiusTrans(lambda m: m ** n, k) for k in range(1, size)]
for n in range(1, 8): print(Jrow(n, 13))
# Alternatively:
from sympy import primefactors as prime_divisors
from fractions import Fraction as QQ
from math import prod as product
def J(n: int, k: int) -> int:
    t = QQ(pow(k, n), 1)
    s = product(1 - QQ(1, pow(p, n)) for p in prime_divisors(k))
    return (t * s).numerator  # the denominator is always 1
for n in range(1, 8): print([J(n, k) for k in range(1, 13)])
# Peter Luschny, Dec 16 2023

J_k(n) = Sum_{d|n} d^k*mu(n/d). - Benoit Cloitre and Michael Orrison (orrison(AT)math.hmc.edu), Jun 07 2002
From Amiram Eldar, Jun 07 2025: (Start)
For a given k, J_k(n) is multiplicative with J_k(p^e) = p^(k*e) - p^(k*e-k).
For a given k, Dirichlet g.f. of J_k(n): zeta(s-k)/zeta(s).
Sum_{i=1..n} J_k(i) ~ n^(k+1) / ((k+1)*zeta(k+1)).
Sum_{n>=1} 1/J_k(n) = Product_{p prime} (1 + p^k/(p^k-1)^2) for k >= 2. (End)

A059380 Array of values of Jordan function J_k(n) read by antidiagonals (version 2).

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 7, 8, 2, 1, 15, 26, 12, 4, 1, 31, 80, 56, 24, 2, 1, 63, 242, 240, 124, 24, 6, 1, 127, 728, 992, 624, 182, 48, 4, 1, 255, 2186, 4032, 3124, 1200, 342, 48, 6, 1, 511, 6560, 16256, 15624, 7502, 2400, 448, 72, 4, 1, 1023, 19682

Offset: 1

N. J. A. Sloane, Jan 28 2001

			Array begins:
1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, ...
1, 3, 8, 12, 24, 24, 48, 48, 72, 72, ...
1, 7, 26, 56, 124, 182, 342, 448, 702, ...
1, 15, 80, 240, 624, 1200, 2400, 3840, ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
R. Sivaramakrishnan, The many facets of Euler's totient. II. Generalizations and analogues, Nieuw Arch. Wisk. (4) 8 (1990), no. 2, 169-187

Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000

See A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5). Columns give A000225, A024023, A020522, A024049, A059387, etc.

Main diagonal gives A067858.

J := proc(n,k) local i,p,t1,t2; t1 := n^k; for p from 1 to n do if isprime(p) and n mod p = 0 then t1 := t1*(1-p^(-k)); fi; od; t1; end;

JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/#]&]/;(n>0)&&IntegerQ[n];
A004736[n_]:=Binomial[Floor[3/2+Sqrt[2*n]],2]-n+1;
A002260[n_]:=n-Binomial[Floor[1/2+Sqrt[2*n]],2];
A059380[n_]:=JordanTotient[A002260[n],A004736[n]]; (* Enrique Pérez Herrero, Dec 19 2010 *)

jordantot(n,k)=sumdiv(n,d,d^k*moebius(n/d));
A002260(n)=n-binomial(floor(1/2+sqrt(2*n)),2);
A004736(n)=binomial(floor(3/2+sqrt(2*n)),2)-n+1;
A059380(n)=jordantot(A002260(n),A004736(n)); \\ Enrique Pérez Herrero, Jan 08 2011

cp's OEIS Frontend

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