A025581 -id:A025581 - cp's OEIS frontend

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

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

A059252 Hilbert's Hamiltonian walk on N X N projected onto x axis: m(3).

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 3, 2, 2, 3, 3, 2, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 2, 2, 3, 3, 3, 2, 2, 3, 4, 5, 5, 4, 4, 4, 5, 5, 6, 6, 7, 7, 7, 6, 6, 7, 7, 7, 6, 6, 5, 4, 4, 5, 5, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 8, 8, 8, 9, 9, 10, 10, 11, 11, 11, 10, 10, 11, 12, 12, 13, 13, 14, 15, 15, 14, 14, 15, 15, 14

Offset: 0

Claude Lenormand (claude.lenormand(AT)free.fr), Jan 23 2001

nonn

This is the X-coordinate of the n-th term in Hilbert's Hamiltonian walk A163359 and the Y-coordinate of its transpose A163357.

			[m(1)=0 0 1 1, m'(1)= 0 1 10] [m(2) =0 0 1 1 2 3 3 2 2 3 3 2 1 1 0 0, m'(2)=0 1 1 0 0 0 1 1 2 2 3 3 3 2 2 3].

Antti Karttunen, Table of n, a(n) for n = 0..65535
Joerg Arndt, Matters Computational (The Fxtbook), section 1.31.1 "The Hilbert curve", page 85, lin2hilbert.
Michael Beeler, R. William Gosper, and Richard Schroeppel, HAKMEM, MIT Artificial Intelligence Laboratory report AIM-239, February 1972. Item 115 by Gosper, page 52. Also HTML transcription. (To use algorithm S or the state table, pad n with high 0-bits to a multiple of 4 bits.)
J. Shallit, Hilbert's spacefilling curve described by automatic, regular, and synchronized sequences, arXiv:2106.01062 [cs.FL], June 2 2021.
Index entries for sequences related to coordinates of 2D curves

See also the y-projection, m'(3), A059253, as well as: A163539, A163540, A163542, A059261, A059285, A163547 and A163529.

void h(unsigned int *x, unsigned int *y, unsigned int l){
x[0] = y[0] = 0; unsigned int *t = NULL; unsigned int n = 0, k = 0;
for(unsigned int i = 1; i>(2*n)){
case 1: x[i] = y[i&k]; y[i] = x[i&k]+(1<Jared Rager, Jan 09 2021 */
(C++) See Fxtbook link.

Initially [m(0) = 0, m'(0) = 0]; recursion: m(2n + 1) = m(2n).m'(2n).f(m'(2n), 2n).c(m(2n), 2n + 1); m'(2n + 1) = m'(2n).f(m(2n), 2n).f(m(2n), 2n).mir(m'(2n)); m(2n) = m(2n - 1).f(m'(2n - 1), 2n - 1).f(m'(2n - 1), 2n - 1).mir(m(2n - 1)); m'(2n) = m'(2n - 1).m(2n - 1).f(m(2n - 1), 2n - 1).c(m'(2n - 1), 2n); where f(m, n) is the alphabetic morphism i := i + 2^n [example: f(0 0 1 1 2 3 3 2 2 3 3 2 1 1 0 0, 2) = 4 4 5 5 6 7 7 6 6 7 7 6 5 5 4 4]; c(m, n) is the complementation to 2^n - 1 alphabetic morphism [example: c(0 0 1 1 2 3 3 2 2 3 3 2 1 1 0 0, 3) = 7 7 6 6 5 4 4 5 5 4 4 5 6 6 7 7]; and mir(m) is the mirror operator [example: mir(0 1 1 0 0 0 1 1 2 2 3 3 3 2 2 3) = 3 2 2 3 3 3 2 2 1 1 0 0 0 1 1 0].

a(n) = A002262(A163358(n)) = A025581(A163360(n)) = A059906(A163356(n)).

Extended by Antti Karttunen, Aug 01 2009

A059253 Hilbert's Hamiltonian walk on N X N projected onto y axis: m'(3).

Original entry on oeis.org

0, 1, 1, 0, 0, 0, 1, 1, 2, 2, 3, 3, 3, 2, 2, 3, 4, 4, 5, 5, 6, 7, 7, 6, 6, 7, 7, 6, 5, 5, 4, 4, 4, 4, 5, 5, 6, 7, 7, 6, 6, 7, 7, 6, 5, 5, 4, 4, 3, 2, 2, 3, 3, 3, 2, 2, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 2, 3, 3, 2, 2, 3, 3, 2, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 2, 2, 3, 3, 3, 2, 2, 3, 4, 5, 5, 4, 4, 4

Offset: 0

Claude Lenormand (claude.lenormand(AT)free.fr), Jan 23 2001

nonn

This is the Y-coordinate of the n-th term in the type I Hilbert's Hamiltonian walk A163359 and the X-coordinate of its transpose A163357.

Antti Karttunen, Table of n, a(n) for n = 0..65535
J. Shallit, Hilbert's spacefilling curve described by automatic, regular, and synchronized sequences, arXiv:2106.01062 [cs.FL], June 2 2021.
Index entries for sequences related to coordinates of 2D curves

See also the y-projection, m(3), A059252 as well as A163538, A163540, A163542, A059261, A059285, A163547 and A163528.

C
```
See A059252.
```

Initially [m(0) = 0, m'(0) = 0]; recursion: m(2n + 1) = m(2n).m'(2n).f(m'(2n), 2n).c(m(2n), 2n + 1); m'(2n + 1) = m'(2n).f(m(2n), 2n).f(m(2n), 2n).mir(m'(2n)); m(2n) = m(2n - 1).f(m'(2n - 1), 2n - 1).f(m'(2n - 1), 2n - 1).mir(m(2n - 1)); m'(2n) = m'(2n - 1).m(2n - 1).f(m(2n - 1), 2n - 1).c(m'(2n - 1), 2n); where f(m, n) is the alphabetic morphism i := i + 2^n [example: f(0 0 1 1 2 3 3 2 2 3 3 2 1 1 0 0, 2) = 4 4 5 5 6 7 7 6 6 7 7 6 5 5 4 4]; c(m, n) is the complementation to 2^n - 1 alphabetic morphism [example: c(0 0 1 1 2 3 3 2 2 3 3 2 1 1 0 0, 3) = 7 7 6 6 5 4 4 5 5 4 4 5 6 6 7 7]; and mir(m) is the mirror operator [example: mir(0 1 1 0 0 0 1 1 2 2 3 3 3 2 2 3) = 3 2 2 3 3 3 2 2 1 1 0 0 0 1 1 0].

a(n) = A025581(A163358(n)) = A002262(A163360(n)) = A059905(A163356(n)).

Extended by Antti Karttunen, Aug 01 2009

A122200 Signature permutations of RIBS-transformations of non-recursive Catalan automorphisms in table A089840.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 4, 3, 2, 1, 0, 5, 4, 3, 2, 1, 0, 6, 5, 4, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2, 1, 0, 8, 8, 6, 5, 4, 3, 2, 1, 0, 9, 7, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 13, 11, 10, 9, 8

Offset: 0

Antti Karttunen, Sep 01 2006

Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th nonrecursive automorphism in the table A089840 with the recursion scheme "RIBS".
In this recursion scheme the given automorphism is applied to all (toplevel) subtrees of the Catalan structure, when it is interpreted as a general tree. Permutations in this table form a countable group, which is isomorphic with the group in A089840. (The RIBS transformation gives the group isomorphism.)
Furthermore, row n of this table is also found as the row A123694(n) in tables A122203 and A122204. If the count of fixed points of the automorphism A089840[n] is given by sequence f, then the count of fixed points of the automorphism A089840[A123694(n)] is given by CONV(f,A000108) (where CONV stands for convolution) and the count of fixed points of the automorphism A122200[n] by INVERT(RIGHT(f)).
The associated Scheme-procedures RIBS and !RIBS can be used to obtain such a transformed automorphism from any constructively or destructively implemented automorphism.
This sequence agrees with A025581 in its initial terms, but then diverges from it. - Antti Karttunen, May 11 2008

Antti Karttunen, paper in preparation, draft available by e-mail.

Index entries for signature-permutations of Catalan automorphisms

Row 0 (identity permutation): A001477, row 1: A122282. See also tables A089840, A122201-A122204, A122283-A122284, A122285-A122288, A122289-A122290.

(define (RIBS foo) (lambda (s) (map foo s)))
(define (!RIBS foo!) (letrec ((bar! (lambda (s) (cond ((pair? s) (foo! (car s)) (bar! (cdr s)))) s))) bar!))

A173787 Triangle read by rows: T(n,k) = 2^n - 2^k, 0 <= k <= n.

Original entry on oeis.org

0, 1, 0, 3, 2, 0, 7, 6, 4, 0, 15, 14, 12, 8, 0, 31, 30, 28, 24, 16, 0, 63, 62, 60, 56, 48, 32, 0, 127, 126, 124, 120, 112, 96, 64, 0, 255, 254, 252, 248, 240, 224, 192, 128, 0, 511, 510, 508, 504, 496, 480, 448, 384, 256, 0, 1023, 1022, 1020, 1016, 1008, 992, 960, 896, 768, 512, 0

Offset: 0

Reinhard Zumkeller, Feb 28 2010

			Triangle begins as:
   0;
   1,  0;
   3,  2,  0;
   7,  6,  4,  0;
  15, 14, 12,  8,  0;
  31, 30, 28, 24, 16, 0;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

[2^n -2^k: k in [0..n], n in [0..15]]; // G. C. Greubel, Jul 13 2021

Table[2^n -2^k, {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 13 2021 *)

flatten([[2^n -2^k for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jul 13 2021

A000120(T(n,k)) = A025581(n,k).
Row sums give A000337.
Central terms give A020522.
T(2*n+1, n) = A006516(n+1).
T(2*n+3, n+2) = A059153(n).
T(n, k) = A140513(n,k) - A173786(n,k), 0 <= k <= n.
T(n, k) = A173786(n,k) - A059268(n+1,k+1), 0 < k <= n.
T(2*n, 2*k) = T(n,k) * A173786(n,k), 0 <= k <= n.
T(n, 0) = A000225(n).
T(n, 1) = A000918(n) for n>0.
T(n, 2) = A028399(n) for n>1.
T(n, 3) = A159741(n-3) for n>3.
T(n, 4) = A175164(n-4) for n>4.
T(n, 5) = A175165(n-5) for n>5.
T(n, 6) = A175166(n-6) for n>6.
T(n, n-4) = A110286(n-4) for n>3.
T(n, n-3) = A005009(n-3) for n>2.
T(n, n-2) = A007283(n-2) for n>1.
T(n, n-1) = A000079(n-1) for n>0.
T(n, n) = A000004(n).

A085207 Array A(x,y): concatenation of binary expansions of x & y, listed antidiagonalwise as A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), ... Zero is expanded as an empty string.

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 3, 5, 6, 3, 4, 7, 10, 7, 4, 5, 9, 14, 11, 12, 5, 6, 11, 18, 15, 20, 13, 6, 7, 13, 22, 19, 28, 21, 14, 7, 8, 15, 26, 23, 36, 29, 22, 15, 8, 9, 17, 30, 27, 44, 37, 30, 23, 24, 9, 10, 19, 34, 31, 52, 45, 38, 31, 40, 25, 10, 11, 21, 38, 35, 60, 53, 46, 39, 56, 41, 26, 11

Offset: 0

Antti Karttunen, Jun 23 2003

			A(2,1) = 5 = '101' in binary, concatenation of 2's binary expansion '10' and 1's '1'. A(1,2) = 6 = '110' in binary, concatenation of '1' and '10'.

Same array in binary: A085209. Transpose: A085208. Variant: A085211. Can be used to compute A085201.

A242378 Square array read by antidiagonals: to obtain A(i,j), replace each prime factor prime(k) in prime factorization of j with prime(k+i).

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 5, 5, 1, 0, 5, 9, 7, 7, 1, 0, 6, 7, 25, 11, 11, 1, 0, 7, 15, 11, 49, 13, 13, 1, 0, 8, 11, 35, 13, 121, 17, 17, 1, 0, 9, 27, 13, 77, 17, 169, 19, 19, 1, 0, 10, 25, 125, 17, 143, 19, 289, 23, 23, 1, 0, 11, 21, 49, 343, 19, 221, 23, 361, 29, 29, 1, 0

Offset: 0

Antti Karttunen, May 12 2014

Each row i is a multiplicative function, being in essence "the i-th power" of A003961, i.e., A(i,j) = A003961^i (j). Zeroth power gives an identity function, A001477, which occurs as the row zero.
The terms in the same column have the same prime signature.
The array is read by antidiagonals: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ... .

			The top-left corner of the array:
  0,   1,   2,   3,   4,   5,   6,   7,   8, ...
  0,   1,   3,   5,   9,   7,  15,  11,  27, ...
  0,   1,   5,   7,  25,  11,  35,  13, 125, ...
  0,   1,   7,  11,  49,  13,  77,  17, 343, ...
  0,   1,  11,  13, 121,  17, 143,  19,1331, ...
  0,   1,  13,  17, 169,  19, 221,  23,2197, ...
...
A(2,6) = A003961(A003961(6)) = p_{1+2} * p_{2+2} = p_3 * p_4 = 5 * 7 = 35, because 6 = 2*3 = p_1 * p_2.

Antti Karttunen, Table of n, a(n) for n = 0..10439; Antidiagonals n = 0..143, flattened

Taking every second column from column 2 onward gives array A246278 which is a permutation of natural numbers larger than 1.
Transpose: A242379.
Row 0: A001477, Row 1: A003961 (from 1 onward), Row 2: A357852 (from 1 onward), Row 3: A045968 (from 7 onward), Row 4: A045970 (from 11 onward).
Column 2: A000040 (primes), Column 3: A065091 (odd primes), Column 4: A001248 (squares of primes), Column 6: A006094 (products of two successive primes), Column 8: A030078 (cubes of primes).
Excluding column 0, a subtable of A297845.
Permutations whose formulas refer to this array: A122111, A241909, A242415, A242419, A246676, A246678, A246684.

A(0,j) = j, A(i,0) = 0, A(i > 0, j > 0) = A003961(A(i-1,j)).

For j > 0, A(i,j) = A297845(A000040(i+1),j) = A297845(j,A000040(i+1)). - Peter Munn, Sep 02 2025

A163528 The X-coordinate of the n-th point in the Peano curve A163334.

Original entry on oeis.org

0, 1, 2, 2, 1, 0, 0, 1, 2, 3, 4, 5, 5, 4, 3, 3, 4, 5, 6, 7, 8, 8, 7, 6, 6, 7, 8, 8, 7, 6, 6, 7, 8, 8, 7, 6, 5, 4, 3, 3, 4, 5, 5, 4, 3, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 3, 4, 5, 5, 4, 3, 3, 4, 5, 6, 7, 8, 8, 7, 6, 6, 7, 8, 9, 10, 11, 11, 10, 9, 9, 10, 11, 12, 13, 14, 14, 13, 12

Offset: 0

Antti Karttunen, Aug 01 2009

nonn

There is a 2-state automaton that accepts exactly those pairs (n,a(n)) where n is represented in base 9 and a(n) in base 3; see accompanying file a163528.pdf - Jeffrey Shallit, Aug 10 2023

Antti Karttunen, Table of n, a(n) for n = 0..59048
Jeffrey Shallit, Automaton for A163528
Index entries for sequences related to coordinates of 2D curves

Cf. A025581, A163335, A002262, A163337, A163325, A163332.

See also: A059252, A059253, A163529, A163530, A163531, A163532.

a(n) = A025581(A163335(n)) = A002262(A163337(n)) = A163325(A163332(n)).

Name corrected by Kevin Ryde, Aug 28 2020

A225630 Array of iterated Landau-like functions, starting maximization of LCM from the partition {1+1+...+1} of n, read downwards antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 4, 6, 2, 1, 1, 1, 6, 12, 6, 2, 1, 1, 1, 6, 30, 12, 6, 2, 1, 1, 1, 12, 30, 60, 12, 6, 2, 1, 1, 1, 15, 84, 60, 60, 12, 6, 2, 1, 1, 1, 20, 120, 420, 60, 60, 12, 6, 2, 1, 1, 1, 30, 180, 840, 420, 60, 60, 12, 6, 2, 1, 1

Offset: 0

Antti Karttunen, May 13 2013

Row 0 consists of all 1's (corresponding to lcm(1,1,...,1) computed from the {1+1+...+1} partition), after which, on each succeeding row, the entry A(row,n) is computed by finding such a partition {p1+p2+...+pk} of n that value of lcm(A(row-1,n),p1,p2,...,pk) is maximized.
This will produce the ordinary Landau's function (A000793) for row 1, the "second order Landau's function" (A225627) for row 2, etc.
For each column n, only finite number of distinct values (A225634(n)) occur, after which the fixed point A003418(n) that has been reached repeats forever.

			Table begins:
  1, 1, 1, 1,  1,  1,  1,   1,   1,  1, ...
  1, 1, 2, 3,  4,  6,  6,  12,  15, 20, ...
  1, 1, 2, 6, 12, 30, 30,  84, 120, ...
  1, 1, 2, 6, 12, 60, 60, 420, 840, ...
  ...

Index entries for sequences related to lcm's

Transpose: A225631. Cf. also A225632, A225634.
Row 0: A000012, row 1: A000793, row 2: A225627, row 3: A225628. Cf. also A225629.
Rows converge towards A003418, which is also the main diagonal of this array.
See A225640 for a variant which uses a similar process, but where the "initial seed" in column n is n instead of 1.

(define (A225630 n) (A225630bi (A025581 n) (A002262 n)))
(define (A225630bi col row) (let ((maxlcm (list 0))) (let loop ((prevmaxlcm 1) (stepsleft row)) (if (zero? stepsleft) prevmaxlcm (begin (gen_partitions col (lambda (p) (set-car! maxlcm (max (car maxlcm) (apply lcm (cons prevmaxlcm p)))))) (loop (car maxlcm) (- stepsleft 1)))))))
(define (gen_partitions m colfun) (let recurse ((m m) (b m) (n 0) (partition (list))) (cond ((zero? m) (colfun partition)) (else (let loop ((i 1)) (recurse (- m i) i (+ 1 n) (cons i partition)) (if (< i (min b m)) (loop (+ 1 i))))))))

A265609 Array read by ascending antidiagonals: A(n,k) the rising factorial, also known as Pochhammer symbol, for n >= 0 and k >= 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 6, 0, 1, 4, 12, 24, 24, 0, 1, 5, 20, 60, 120, 120, 0, 1, 6, 30, 120, 360, 720, 720, 0, 1, 7, 42, 210, 840, 2520, 5040, 5040, 0, 1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320, 0

Offset: 0

Peter Luschny, Dec 19 2015

The Pochhammer function is defined P(x,n) = x*(x+1)*...*(x+n-1). By convention P(0,0) = 1.
From Antti Karttunen, Dec 19 2015: (Start)
Apart from the initial row of zeros, if we discard the leftmost column and divide the rest of terms A(n,k) with (n+k) [where k is now the once-decremented column index of the new, shifted position] we get the same array back. See the given recursive formula.
When the numbers in array are viewed in factorial base (A007623), certain repeating patterns can be discerned, at least in a few of the topmost rows. See comment in A001710 and arrays A265890, A265892. (End)
A(n,k) is the k-th moment (about 0) of a gamma (Erlang) distribution with shape parameter n and rate parameter 1. - Geoffrey Critzer, Dec 24 2018

			Square array A(n,k) [where n=row, k=column] is read by ascending antidiagonals as:
A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), A(3,0), A(2,1), A(1,2), A(0,3), ...
Array starts:
n\k [0  1   2    3     4      5        6         7          8]
--------------------------------------------------------------
[0] [1, 0,  0,   0,    0,     0,       0,        0,         0]
[1] [1, 1,  2,   6,   24,   120,     720,     5040,     40320]
[2] [1, 2,  6,  24,  120,   720,    5040,    40320,    362880]
[3] [1, 3, 12,  60,  360,  2520,   20160,   181440,   1814400]
[4] [1, 4, 20, 120,  840,  6720,   60480,   604800,   6652800]
[5] [1, 5, 30, 210, 1680, 15120,  151200,  1663200,  19958400]
[6] [1, 6, 42, 336, 3024, 30240,  332640,  3991680,  51891840]
[7] [1, 7, 56, 504, 5040, 55440,  665280,  8648640, 121080960]
[8] [1, 8, 72, 720, 7920, 95040, 1235520, 17297280, 259459200]
.
Seen as a triangle, T(n, k) = Pochhammer(n - k, k), the first few rows are:
   [0] 1;
   [1] 1, 0;
   [2] 1, 1,  0;
   [3] 1, 2,  2,   0;
   [4] 1, 3,  6,   6,    0;
   [5] 1, 4, 12,  24,   24,    0;
   [6] 1, 5, 20,  60,  120,  120,     0;
   [7] 1, 6, 30, 120,  360,  720,   720,     0;
   [8] 1, 7, 42, 210,  840, 2520,  5040,  5040,     0;
   [9] 1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320, 0.

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, Addison-Wesley, 1994.
H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 355.

NIST Digital Library of Mathematical Functions, Pochhammer's Symbol
Index entries for sequences related to factorial numbers

Triangle giving terms only up to column k=n: A124320.
Row 0: A000007, row 1: A000142, row 3: A001710 (from k=1 onward, shifted two terms left).
Column 0: A000012, column 1: A001477, column 2: A002378, columns 3-7: A007531, A052762, A052787, A053625, A159083 (shifted 2 .. 6 terms left respectively, i.e. without the extra initial zeros), column 8: A239035.
Row sums of the triangle: A000522.
A(n, n) = A000407(n-1) for n>0.
2^n*A(1/2,n) = A001147(n).
Cf. also A007623, A008279 (falling factorial), A173333, A257505, A265890, A265892.

for n from 0 to 8 do seq(pochhammer(n,k), k=0..8) od;

Table[Pochhammer[n, k], {n, 0, 8}, {k, 0, 8}]

for n in (0..8): print([rising_factorial(n,k) for k in (0..8)])

(define (A265609 n) (A265609bi (A025581 n) (A002262 n)))
(define (A265609bi row col) (if (zero? col) 1 (* (+ row col -1) (A265609bi row (- col 1)))))
;; Antti Karttunen, Dec 19 2015

A(n,k) = Gamma(n+k)/Gamma(n) for n > 0 and n^k for n=0.
A(n,k) = Sum_{j=0..k} n^j*S1(k,j), S1(n,k) the Stirling cycle numbers A132393(n,k).
A(n,k) = (k-1)!/(Sum_{j=0..k-1} (-1)^j*binomial(k-1, j)/(j+n)) for n >= 1, k >= 1.
A(n,k) = (n+k-1)*A(n,k-1) for k >= 1, A(n,0) = 1. - Antti Karttunen, Dec 19 2015
E.g.f. for row k: 1/(1-x)^k. - Geoffrey Critzer, Dec 24 2018
A(n, k) = FallingFactorial(n + k - 1, k). - Peter Luschny, Mar 22 2022
G.f. for row n as a continued fraction of Stieltjes type: 1/(1 - n*x/(1 - x/(1 - (n+1)*x/(1 - 2*x/(1 - (n+2)*x/(1 - 3*x/(1 - ... ))))))). See Wall, Chapter XVIII, equation 92.5. Cf. A226513. - Peter Bala, Aug 27 2023

cp's OEIS Frontend

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

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A059252 Hilbert's Hamiltonian walk on N X N projected onto x axis: m(3).

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A059253 Hilbert's Hamiltonian walk on N X N projected onto y axis: m'(3).

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A122200 Signature permutations of RIBS-transformations of non-recursive Catalan automorphisms in table A089840.

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A173787 Triangle read by rows: T(n,k) = 2^n - 2^k, 0 <= k <= n.

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A085207 Array A(x,y): concatenation of binary expansions of x & y, listed antidiagonalwise as A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), ... Zero is expanded as an empty string.

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A242378 Square array read by antidiagonals: to obtain A(i,j), replace each prime factor prime(k) in prime factorization of j with prime(k+i).

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A163528 The X-coordinate of the n-th point in the Peano curve A163334.

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