A360173 -id:A360173 - cp's OEIS frontend

A360229 Row sums of triangle A360173.

0, 1, 3, 6, 16, 36, 73, 156, 324, 677, 1405, 2864, 5906, 12058, 24548, 49928, 101434, 206173, 417141, 844256, 1707622, 3452998, 6970196, 14058528, 28368774, 57197983, 115239846, 232020596, 467296470, 940684267, 1892396396, 3805806218, 7654402454, 15391563411

Offset: 0

John Tyler Rascoe, Jan 30 2023

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..800

Cf. A141001, A141002, A360173.

b:= proc(n) option remember; `if`(n=0, 1, (p->
       add((t-> `if`(i
      add(i*coeff(p, x, i), i=0..degree(p)))(b(n))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Jan 30 2023

b[n_] := b[n] = If[n == 0, 1, Function[p, Sum[Function[t, If[iJean-François Alcover, Apr 24 2025, after Alois P. Heinz *)

A365968 Irregular triangle read by rows: T(n,k) (0 <= n, 0 <= k < 2^n). An infinite binary tree with root node 0 in row n = 0. Each node then has left child (2j) - k - 1 and right child (2j) - k + 1, where j and k are the values of the parent and grandparent nodes respectively.

Original entry on oeis.org

0, -1, 1, -3, -1, 1, 3, -6, -4, -2, 0, 0, 2, 4, 6, -10, -8, -6, -4, -4, -2, 0, 2, -2, 0, 2, 4, 4, 6, 8, 10, -15, -13, -11, -9, -9, -7, -5, -3, -7, -5, -3, -1, -1, 1, 3, 5, -5, -3, -1, 1, 1, 3, 5, 7, 3, 5, 7, 9, 9, 11, 13, 15, -21, -19, -17, -15, -15, -13, -11, -9

Offset: 0

John Tyler Rascoe, Sep 23 2023

For n in A014601 row n will contain all even numbers from 0 to A000217(n).

For n in A042963 row n will contain all odd numbers from 1 to A000217(n).

			Triangle begins:
        k=0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
  n=0:    0;
  n=1:   -1,  1;
  n=2:   -3, -1,  1,  3;
  n=3:   -6, -4, -2,  0,  0,  2,  4,  6;
  n=4:  -10, -8, -6, -4, -4, -2,  0,  2, -2,  0,  2,  4,  4,  6,  8, 10;
  ...
The binary tree starts with root 0 in row n = 0. For rows n < 2, k = 0.
In row n = 3, the parent node -3 has left child -6 = 2*(-3) - (-1) - 1.
The tree begins:
row
[n]
[0]                   ______0______
                     /             \
[1]              __-1__           __1__
                /      \         /     \
[2]           -3       -1       1       3
              / \      / \     / \     / \
[3]         -6  -4   -2   0   0   2   4   6
.

John Tyler Rascoe, Rows n = 0..12, flattened

Cf. A004718, A274575, A360173, A363718.

T(n,k) = sum(i=0,n-1, if(bittest(k,i), i+1, -(i+1))); \\ Kevin Ryde, Nov 14 2023

def A365968(n, k):
    b, x = bin(k)[2:].zfill(n), 0
    for i in range(0, n):
        x += (-1)**(int(b[n-(i+1)])+1)*(i+1)
    return(x) # John Tyler Rascoe, Nov 12 2023

T(n,k) = - Sum_{i=0..n-1} (i+1)*(-1)^b[i] where the binary expansion of k is k = Sum_{i=0..n-1} b[i]*2^i. - Kevin Ryde, Nov 14 2023

A360298 Irregular triangle (an infinite binary tree) read by rows. The tree has root node 1 in row n = 1. For n > 1, each node with value m in row n-1 has a left child with value m / n if n divides m, and a right child with value m * n.

Original entry on oeis.org

1, 2, 6, 24, 120, 20, 720, 140, 5040, 1120, 630, 40320, 10080, 70, 5670, 4480, 362880, 1008, 100800, 7, 700, 567, 56700, 448, 44800, 36288, 3628800, 11088, 1108800, 77, 7700, 6237, 623700, 4928, 492800, 399168, 39916800, 924, 133056, 92400, 13305600, 924, 92400, 74844, 51975, 7484400, 59136, 5913600, 33264, 4790016, 3326400, 479001600

Offset: 1

Rémy Sigrist, Feb 02 2023

This sequence is a variant of A360173; here we use divisions and multiplications, there subtractions and additions.

The n-th row has A360299(n) terms, starts with A008336(n+1) and ends with A000142(n).

			The tree begins:
  n     n-th row
  --    --------
   1    1___
            |
   2        2___
                |
   3            6___
                    |
   4               24___
                        |
   5      _____________120_____________
         |                             |
   6    20___                         720___
             |                              |
   7        140___                   _____5040_____
                  |                 |              |
   8            1120__           __630__       __40320__
                      |         |       |     |         |
   9                10080      70     5670  4480     362880

Rémy Sigrist, Table of n, a(n) for n = 1..11270 (rows for n = 1..26 flattened)

Cf. A000142, A008336, A360173, A360299 (row lengths), A360300.

row(n) = { my (r = [1]); for (h = 2, n, r=concat(apply(v -> if (v%h==0, [v/h, v*h], [v*h]), r))); return (r) }

from functools import cache
@cache
def row(n):
    if n == 1: return [1]
    out = []
    for r in row(n-1): out += ([r//n] if r%n == 0 else []) + [r*n]
    return out
print([an for r in range(1, 13) for an in row(r)]) # Michael S. Branicky, Feb 02 2023

T(n, 1) = A008336(n+1).
T(n, A360299(n)) = A000142(n).
T(p, k) = p * T(p-1, k) for any prime number p.

A360255 Irregular triangle (an infinite binary tree) read by rows: see Comments section for definition.

Original entry on oeis.org

0, 1, 3, 6, 2, 10, 7, 5, 15, 13, 11, 9, 21, 20, 4, 18, 2, 16, 14, 28, 12, 28, 12, 26, 8, 24, 22, 20, 36, 21, 19, 37, 21, 17, 35, 17, 33, 13, 31, 11, 29, 27, 45, 11, 31, 9, 29, 27, 47, 31, 7, 27, 25, 45, 7, 27, 23, 43, 23, 41, 19, 39, 17, 37, 35, 55, 22, 42, 18

Offset: 0

John Tyler Rascoe, Jan 30 2023

The binary tree has root node 0, in row n=0. The left child is m - n and the right child is m + n, where m is the parent node and n is the row of the child. A given node will only have a child if the child is nonnegative and the value of the child is not present in the path from the parent to the root, including the root value itself.

The n-th row will have A321535(n) nodes. The rightmost border is A000217.

			The binary tree starts with root 0 in row n = 0. In row n = 3, the parent node m = 3 does not have a left child since 3 - 3 = 0 is included in the path from the parent to the root {3,1,0}.
The tree begins:
row
[n]
[0]           0
               \
[1]             1
                 \
[2]               3
                   \
[3]               __6__
                 /     \
[4]             2      10
                 \    /  \
[5]               7  5    15

Rémy Sigrist, Table of n, a(n) for n = 0..9517 (rows for n = 0..21 flattened)

Cf. A000217, A141001, A141002, A321535, A360173.

function a = A360255( max_row )
    p = 0; a = 0; pos = 1;
    for n = 1:max_row
        for k = pos:length(a)
            h =[]; o = p(k);
            while o > 0
                h = [h a(o)]; o = p(o);
            end
            if a(k)-n > 0
                if isempty(find(h == a(k)-n, 1))
                    p = [p k]; a = [a a(k)-n];
                end
            end
            if isempty(find(h == a(k)+n, 1))
                p = [p k]; a = [a a(k)+n];
            end
        end
        pos = k+1;
    end
end % Thomas Scheuerle, Jan 31 2023

cp's OEIS Frontend

A360229 Row sums of triangle A360173.

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Maple

Mathematica

A365968 Irregular triangle read by rows: T(n,k) (0 <= n, 0 <= k < 2^n). An infinite binary tree with root node 0 in row n = 0. Each node then has left child (2*j) - k - 1 and right child (2*j) - k + 1, where j and k are the values of the parent and grandparent nodes respectively.

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PARI

Python

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A360298 Irregular triangle (an infinite binary tree) read by rows. The tree has root node 1 in row n = 1. For n > 1, each node with value m in row n-1 has a left child with value m / n if n divides m, and a right child with value m * n.

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PARI

Python

Formula

A360255 Irregular triangle (an infinite binary tree) read by rows: see Comments section for definition.

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MATLAB

A365968 Irregular triangle read by rows: T(n,k) (0 <= n, 0 <= k < 2^n). An infinite binary tree with root node 0 in row n = 0. Each node then has left child (2j) - k - 1 and right child (2j) - k + 1, where j and k are the values of the parent and grandparent nodes respectively.