A070909 -id:A070909 - cp's OEIS frontend

A177990 Triangle read by rows, variant of A070909, a cellular automaton.

1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1

Offset: 1

Gary W. Adamson, May 16 2010

A070909 * A177990 = A177991
A007318 * A177990 = A177992
A177990 * A007318 = A177993

			First few rows of the triangle =
1;
0, 1;
0, 1, 1;
0, 1, 0, 1;
0, 1, 0, 1, 1;
0, 1, 0, 1, 0, 1;
0, 1, 0, 1, 0, 1, 1;
0, 1, 0, 1, 0, 1, 0, 1;
0, 1, 0, 1, 0, 1, 0, 1, 1;
0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1;
0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
...

Cf. A070909, A177991, A177992, A177993.

rows = 10; ca = CellularAutomaton[28, {{1}, 0}, rows]; Table[ca[[k, 2 ;; k]], {k, 2, rows+1}] // Flatten (* Jean-François Alcover, Jul 12 2017 *)

Infinite lower triangular matrix, variant of A070909: Columns alternate between (1,0,0,0,...) and (1,1,1,...); whereas A070909 leftmost column = (1,1,1,...).

A177994 Triangle read by rows, A177990 * A070909.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1, 1

Offset: 0

Gary W. Adamson, May 16 2010

Row sums = A101881: (1, 2, 4, 5, 8, 9, 13, 14,...).

Double Riordan array ( 1/((1 - x)*(1 - x^2)); x*(1 - x^2), x/(1 - x^2) ) as defined in Davenport et al. The set of double Riordan arrays of the form ( g(x); x*f_1(x), x*f_2(x) ), where f_1(x)*f_2(x) = 1, forms a group under matrix multiplication. Here g, f_1 and f_2 denote power series with constant term equal to 1. This is the array (( 1/((1 - x)*(1 - x^2)), 1/(1 - x) )) in the notation of the Bala link. - Peter Bala, Aug 26 2021

			First few rows of the triangle =
1;
1, 1;
2, 1, 1;
2, 1, 1, 1;
3, 1, 2, 1, 1;
3, 1, 2, 1, 1, 1;
4, 1, 3, 1, 2, 1, 1;
4, 1, 3, 1, 2, 1, 1, 1;
5, 1, 4, 1, 3, 1, 2, 1, 1;
5, 1, 4, 1, 3, 1, 2, 1, 1, 1;
6, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1;
6, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1, 1;
7, 1, 6, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1;
7, 1, 6, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1, 1;
...

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
Peter Bala, Matrices with repeated columns - the generalised Appell groups
D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012).

Cf. A177991 = A070909 * A177990.

Cf. A101881.

a177994 n k = a177994_tabl !! n !! k
a177994_row n = a177994_tabl !! n
a177994_tabl = [1] : [1,1] : map f a177994_tabl
               where f xs@(x:_) = (x + 1) : 1 : xs
-- Reinhard Zumkeller, Feb 20 2015

As infinite lower triangular matrices, A177990 * A070909

A177991 Triangle read by rows, A070909 * A177990.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1, 1

Offset: 0

Gary W. Adamson, May 16 2010

Row sums = A117142: (1, 2, 3, 5, 6, 9,...)

			First few rows of the triangle =
1;
1, 1;
1, 1, 1;
1, 2, 1, 1;
1, 2, 1, 1, 1;
1, 3, 1, 2, 1, 1;
1, 3, 1, 2, 1, 1, 1;
1, 4, 1, 3, 1, 2, 1, 1;
1, 4, 1, 3, 1, 2, 1, 1, 1;
1, 5, 1, 4, 1, 3, 1, 2, 1, 1;
1, 5, 1, 4, 1, 3, 1, 2, 1, 1, 1;
1, 6, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1;
1, 6, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1, 1;
1, 7, 1, 6, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1;
1, 7, 1, 6, 1, 5, 1, 4, 1, 3, 1, 2, 1, 1, 1;
...

Peter Bala, Matrices with repeated columns - the generalised Appell groups

A177994 = A177990 * A070909.

Cf. A117142.

As infinite lower triangular matrices, A070909 * A177990

a(28)=1 inserted before the first "4" by Georg Fischer, Dec 28 2021

A177954 Triangle read by rows, A070909 * Pascal's triangle.

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 3, 5, 4, 1, 3, 6, 7, 4, 1, 4, 11, 17, 14, 6, 1, 4, 12, 22, 24, 16, 6, 1, 5, 19, 43, 59, 51, 27, 8, 1, 5, 20, 50, 80, 86, 62, 29, 8, 1, 6, 29, 86, 164, 212, 188, 113, 44, 10, 1

Offset: 0

Gary W. Adamson, May 15 2010

Row sums = A086893: (1, 3, 5, 13, 21, 53, 85,...).

			First few rows of the triangle =
1;
2, 1;
2, 2, 1;
3, 5, 4, 1;
3, 6, 7, 4, 1;
4, 11, 17, 14, 6, 1;
4, 12, 22, 24, 16, 6, 1;
5, 19, 43, 59, 51, 27, 8, 1;
5, 20, 50, 80, 86, 62, 29, 8, 1;
6, 29, 86, 164, 212, 188, 113, 44, 10, 1;
6, 30, 95, 200, 296, 314, 239, 128, 46, 10, 1;
7, 41, 150, 365, 626, 776, 701, 458, 211, 65, 12, 1;
...

Cf. A070909, A086893

Triangle read by rows, A070909 * A007318

a(46) corrected by Georg Fischer, May 20 2022

A181650 Inverse of number triangle A070909.

Original entry on oeis.org

1, -1, 1, -1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, 1, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1

Offset: 0

Paul Barry, Nov 03 2010

Generalized (conditional) Riordan array with k-th column generated by x^k*(1-x-x^2) if k is even, x^k otherwise.
Triangle T(n,k), read by rows, given by (-1,2,-1/2,-1/2,0,0,0,0,0,0,0,...) DELTA (1,0,-1,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 19 2011
Double Riordan array (1 - x - x^2; x/(1 - x - x^2), x*(1 - x - x^2)) as defined in Davenport et al. - Peter Bala, Aug 15 2021

			Triangle begins
   1,
  -1,  1,
  -1,  0,  1,
   0,  0, -1,  1,
   0,  0, -1,  0,  1,
   0,  0,  0,  0, -1,  1,
   0,  0,  0,  0, -1,  0,  1,
   0,  0,  0,  0,  0,  0, -1,  1,
   0,  0,  0,  0,  0,  0, -1,  0,  1,
   0,  0,  0,  0,  0,  0,  0,  0, -1,  1,
   0,  0,  0,  0,  0,  0,  0,  0, -1,  0,  1
Production matrix begins
  -1,  1,
  -2,  1,  1,
  -1,  1, -1,  1,
  -1,  1, -2,  1,  1,
  -1,  1, -1,  1, -1,  1,
  -1,  1, -1,  1, -2,  1,  1,
  -1,  1, -1,  1, -1,  1, -1,  1,
  -1,  1, -1,  1, -1,  1, -2,  1,  1,
  -1,  1, -1,  1, -1,  1, -1,  1, -1,  1

D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012).

Cf. A084221, A084938.

G.f.: (1+(y-1)*x-x^2)/((1-y*x)*(1+y*x)). - Philippe Deléham, Nov 19 2011

A181651 Eigentriangle of number triangle A070909.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 6, 2, 2, 1, 1, 9, 3, 3, 1, 1, 1, 18, 6, 6, 2, 2, 1, 1, 27, 9, 9, 3, 3, 1, 1, 1, 54, 18, 18, 6, 6, 2, 2, 1, 1, 81, 27, 27, 9, 9, 3, 3, 1, 1, 1, 162, 54, 54, 18, 18, 6, 6, 2, 2, 1, 1

Offset: 0

Paul Barry, Nov 03 2010

First column is (essentially) A038754. Row sums are A068911. Inverse is A181652.
Generalized (conditional) Riordan array with k-th column generated by
  x^k*(1+x-x^2)/(1-3x^2) if k is even,
  (1+x-2x^2-x^3)/(1-3x^2) if k is odd.

			Triangle begins
    1;
    1,  1;
    2,  1,  1;
    3,  1,  1,  1;
    6,  2,  2,  1,  1;
    9,  3,  3,  1,  1,  1;
   18,  6,  6,  2,  2,  1,  1;
   27,  9,  9,  3,  3,  1,  1,  1;
   54, 18, 18,  6,  6,  2,  2,  1,  1;
   81, 27, 27,  9,  9,  3,  3,  1,  1,  1;
  162, 54, 54, 18, 18,  6,  6,  2,  2,  1,  1;

Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490 [math.CO], 2011.

Cf. A038754, A068911, A070909, A181652.

A177953 Triangle read by rows, binomial transform of triangle A070909.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 8, 3, 4, 1, 16, 4, 11, 4, 1, 32, 5, 26, 10, 6, 1, 64, 6, 57, 20, 22, 6, 1, 128, 7, 120, 35, 64, 21, 8, 1, 256, 8, 247, 56, 163, 56, 37, 8, 1, 512, 9, 502, 84, 382, 126, 130, 36, 10, 1, 1024, 10, 1013, 120, 848, 252, 386, 120, 56, 10, 1, 2048, 11, 2036, 165, 1816, 462, 1024, 330, 232, 55, 12, 1

Offset: 0

Gary W. Adamson, May 15 2010

Row sums = A045891 starting with offset 1: (1, 3, 7, 16, 36,...).

			First few rows of the triangle =
1;
2, 1;
4, 2, 1;
8, 3, 4, 1;
16, 4, 11, 4, 1;
32, 5, 26, 10, 6, 1;
64, 6, 57, 20, 22, 6, 1;
128, 7, 120, 35, 64, 21, 8, 1;
256, 8, 247, 56, 163, 56, 37, 8, 1;
512, 9, 502, 84, 382, 126, 130, 36, 10, 1;
1024, 10, 1013, 120, 848, 252, 386, 120, 56, 10, 1;
...

Cf. A045891, A070909.

Triangle read by rows, A007318 * A070909.

a(27) = 1 inserted and more terms from Georg Fischer, May 29 2023

A178650 Triangle read by rows A051731 * A070909.

Original entry on oeis.org

1, 2, 1, 2, 0, 1, 3, 1, 1, 1, 2, 0, 1, 0, 1, 4, 1, 2, 0, 1, 1, 2, 0, 1, 0, 1, 0, 1, 4, 1, 2, 1, 1, 0, 1, 1, 3, 0, 2, 0, 1, 0, 1, 0, 1, 4, 1, 2, 0, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 6, 1, 4, 1, 2, 1, 1, 0, 1, 0, 1, 1

Offset: 1

Gary W. Adamson, May 31 2010

Row sums = A069734: (1, 3, 3, 6, 4, 9, 5, 11,...)

Left border = d(n), A000005

			First few rows of the triangle =
1;
2, 1;
2, 0, 1;
3, 1, 1, 1;
2, 0, 1, 0, 1;
4, 1, 2, 0, 1, 1;
2, 0, 1, 0, 1, 0, 1;
4, 1, 2, 1, 1, 0, 1, 1;
3, 0, 2, 0, 1, 0, 1, 0, 1;
4, 1, 2, 0, 2, 0, 1, 0, 1, 1;
2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
6, 1, 4, 1, 2, 1, 1, 0, 1, 0, 1, 1;
2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
4, 1, 2, 0, 2, 0, 2, 0, 1, 0, 1, 0, 1, 1;
4, 0, 3, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1
5, 1, 3, 1, 2, 0, 2, 1, 1, 0, 1, 0, 1, 0, 1, 1;
2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
...

Cf. A051731, A070909, A069734.

As infinite lower triangular matrices, A051731 * A070909; where A051731 = the inverse Mobius transform.

A038754 a(2n) = 3^n, a(2n+1) = 2*3^n.

Original entry on oeis.org

1, 2, 3, 6, 9, 18, 27, 54, 81, 162, 243, 486, 729, 1458, 2187, 4374, 6561, 13122, 19683, 39366, 59049, 118098, 177147, 354294, 531441, 1062882, 1594323, 3188646, 4782969, 9565938, 14348907, 28697814, 43046721, 86093442, 129140163, 258280326, 387420489

Offset: 0

Henry Bottomley, May 03 2000

In general, for the recurrence a(n) = a(n-1)*a(n-2)/a(n-3), all terms are integers iff a(0) divides a(2) and first three terms are positive integers, since a(2n+k) = a(k)*(a(2)/a(0))^n for all nonnegative integers n and k.
Equals eigensequence of triangle A070909; (1, 1, 2, 3, 6, 9, 18, ...) shifts to the left with multiplication by triangle A070909. - Gary W. Adamson, May 15 2010
The a(n) represent all paths of length (n+1), n >= 0, starting at the initial node on the path graph P_5, see the second Maple program. - Johannes W. Meijer, May 29 2010
a(n) is the difference between numbers of multiple of 3 evil (A001969) and odious (A000069) numbers in interval [0, 2^(n+1)). - Vladimir Shevelev, May 16 2012
A "half-geometric progression": to obtain a term (beginning with the third one) we multiply the before previous one by 3. - Vladimir Shevelev, May 21 2012
Pisano periods: 1, 2, 1, 4, 8, 2, 12, 4, 1, 8, 10, 4, 6, 12, 8, 8, 32, 2, 36, 8, ... . - R. J. Mathar, Aug 10 2012
Numbers k such that the k-th cyclotomic polynomial has a root mod 3. - Eric M. Schmidt, Jul 31 2013
Range of row n of the circular Pascal array of order 6. - Shaun V. Ault, Jun 05 2014
Also, the number of walks of length n on the graph 0--1--2--3--4 starting at vertex 1. - Sean A. Irvine, Jun 03 2025

			In the interval [0,2^5) we have 11 multiples of 3 numbers, from which 10 are evil and only one (21) is odious. Thus a(4) = 10 - 1 = 9. - _Vladimir Shevelev_, May 16 2012

Indranil Ghosh, Table of n, a(n) for n = 0..1500 (first 401 terms from T. D. Noe)
S. V. Ault and C. Kicey, Counting paths in corridors using circular Pascal arrays, Discrete Mathematics, Vol. 332 (2014), pp. 45-54.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Nachum Dershowitz, Between Broadway and the Hudson, arXiv:2006.06516 [math.CO], 2020.
Sean A. Irvine, Walks on Graphs.
Richard L. Ollerton and Anthony G. Shannon, Some properties of generalized Pascal squares and triangles, Fib. Q., 36 (1998), 98-109. See pages 106-7.
Vladimir Shevelev, On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m, arXiv:0710.3177 [math.NT], 2007.
M. B. Wells, Elements of Combinatorial Computing, Pergamon, Oxford, 1971. [Annotated scanned copy of pages 237-240]
Index entries for linear recurrences with constant coefficients, signature (0,3).

Cf. A000045, A000069, A000079, A000120, A000244, A001969, A006720, A006721.
Cf. A006722, A006723, A008776, A028495, A030436, A037124, A048328, A061551.
Cf. A068911, A070909, A087503, A124302, A128588, A133464, A140730, A140740.
Cf. A178381, A182751, A182752, A182753, A182754, A182755, A182756, A182757.
Fifth row of the array A094718.

import Data.List (transpose)
a038754 n = a038754_list !! n
a038754_list = concat $ transpose [a000244_list, a008776_list]
-- Reinhard Zumkeller, Oct 19 2015

[n le 2 select n else 3*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Aug 18 2016

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=3*a[n-2]+2 od: seq(a[n]+1, n=0..34); # Zerinvary Lajos, Mar 20 2008
with(GraphTheory): P:=5: G:=PathGraph(P): A:= AdjacencyMatrix(G): nmax:=35; for n from 1 to nmax do B(n):=A^n; a(n):=add(B(n)[1,k],k=1..P) od: seq(a(n),n=1..nmax); # Johannes W. Meijer, May 29 2010

LinearRecurrence[{0,3},{1,2},40] (* Harvey P. Dale, Jan 26 2014 *)
CoefficientList[Series[(1+2x)/(1-3x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 18 2016 *)
Module[{nn=20,c},c=3^Range[0,nn];Riffle[c,2c]] (* Harvey P. Dale, Aug 21 2021 *)

PARI
```
a(n)=(1/6)*(5-(-1)^n)*3^floor(n/2)
```
PARI
```
a(n)=3^(n>>1)<
				
```

[2^(n%2)*3^((n-(n%2))/2) for n in range(61)] # G. C. Greubel, Oct 10 2022

a(n) = a(n-1)*a(n-2)/a(n-3) with a(0)=1, a(1)=2, a(2)=3.
a(2*n) = (3/2)*a(2*n-1) = 3^n, a(2*n+1) = 2*a(2*n) = 2*3^n.
From Benoit Cloitre, Apr 27 2003: (Start)
a(1)=1, a(n)= 2*a(n-1) if a(n-1) is odd, or a(n)= (3/2)*a(n-1) if a(n-1) is even.
a(n) = (1/6)*(5-(-1)^n)*3^floor(n/2).
a(2*n) = a(2*n-1) + a(2*n-2) + a(2*n-3).
a(2*n+1) = a(2*n) + a(2*n-1). (End)
G.f.: (1+2*x)/(1-3*x^2). - Paul Barry, Aug 25 2003
From Reinhard Zumkeller, Sep 11 2003: (Start)
a(n) = (1 + n mod 2) * 3^floor(n/2).
a(n) = A087503(n) - A087503(n-1). (End)
a(n) = sqrt(3)*(2+sqrt(3))*(sqrt(3))^n/6 - sqrt(3)*(2-sqrt(3))*(-sqrt(3))^n/6. - Paul Barry, Sep 16 2003
From Reinhard Zumkeller, May 26 2008: (Start)
a(n) = A140740(n+2,2).
a(n+1) = a(n) + a(n - n mod 2). (End)
If p(i) = Fibonacci(i-3) and if A is the Hessenberg matrix of order n defined by A(i,j) = p(j-i+1), (i<=j), A(i,j)=-1, (i=j+1), and A(i,j)=0 otherwise. Then, for n>=1, a(n-1) = (-1)^n det A. - Milan Janjic, May 08 2010
a(n) = A182751(n) for n >= 2. - Jaroslav Krizek, Nov 27 2010
a(n) = Sum_{i=0..2^(n+1), i==0 (mod 3)} (-1)^A000120(i). - Vladimir Shevelev, May 16 2012
a(0)=1, a(1)=2, for n>=3, a(n)=3*a(n-2). - Vladimir Shevelev, May 21 2012
Sum_(n>=0) 1/a(n) = 9/4. - Alexander R. Povolotsky, Aug 24 2012
a(n) = sqrt(3*a(n-1)^2 + (-3)^(n-1)). - Richard R. Forberg, Sep 04 2013
a(n) = 2^((1-(-1)^n)/2)*3^((2*n-1+(-1)^n)/4). - Luce ETIENNE, Aug 11 2014
From Reinhard Zumkeller, Oct 19 2015: (Start)
a(2*n) = A000244(n), a(2*n+1) = A008776(n).
For n > 0: a(n+1) = a(n) + if a(n) odd then min{a(n), a(n-1)} else max{a(n), a(n-1)}, see also A128588. (End)
E.g.f.: (7*cosh(sqrt(3)*x) + 4*sqrt(3)*sinh(sqrt(3)*x) - 4)/3. - Stefano Spezia, Feb 17 2022
Sum_{n>=0} (-1)^n/a(n) = 3/4. - Amiram Eldar, Dec 02 2022

A169623 Generalized Pascal triangle read by rows: T(n,0) = T(0,n) = 1 for n >= 0, T(n,k) = 0 for k < 0 or k > n; otherwise T(n,k) = T(n-2,k-2) + T(n-2,k-1) + T(n-2,k) for 1 <= k <= n-1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 3, 5, 5, 3, 1, 1, 3, 6, 7, 6, 3, 1, 1, 4, 9, 13, 13, 9, 4, 1, 1, 4, 10, 16, 19, 16, 10, 4, 1, 1, 5, 14, 26, 35, 35, 26, 14, 5, 1, 1, 5, 15, 30, 45, 51, 45, 30, 15, 5, 1, 1, 6, 20, 45, 75, 96, 96, 75, 45, 20, 6, 1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Dec 03 2009

The borders are all 1's, with zero entries outside. To get an internal entry, use the rule that D = A+B+C here:
         A   B   C
       *   *   *   *
     *   *   D   *   *
That is, add the three terms directly above you, two rows back.
This is the triangle er(n,k) defined in the Ehrenborg and Readdy link. See Proposition 2.4 and Table 1. - Michel Marcus, Sep 14 2016
If the offset is changed from 0 to 1, this is also the table U(n,k) of the coefficients [x^k] p_n(x) of the polynomials p_n(x) = (x + 1)*p_{n-1}(x) (if n even), p_n = (x^2 + x + 1)^floor(n/2) if n odd.
May be split into two triangles by taking the even-numbered and odd-numbered rows separately: the even-numbered rows give A027907.
From Peter Bala, Aug 19 2021: (Start)
Let M denote the lower unit triangular array A070909. For k = 0,1,2,..., define M(k) to be the lower unit triangular block array
  /I_k 0\
  \ 0  M/
  having the k x k identity matrix I_k as the upper left block; in particular, M(0) = M. Then the present triangle equals the infinite matrix product M(0)*M(1)*M(2)*... (which is clearly well-defined). See the Example section below. The proof uses the hockey-stick identities from the Formula section. (End)

			Triangle begins:
                    1
                  1   1
                1   1   1
              1   2   2   1
            1   2   3   2   1
          1   3   5   5   3   1
        1   3   6   7   6   3   1
      1   4   9  13  13   9   4   1
    1   4  10  16  19  16  10   4   1
  ...
As a square array read by antidiagonals:
  1, 1, 1,  1,  1,  1,  1,  1,  1,  1,  1,  1, 1, 1, 1, 1, ...
  1, 1, 2,  2,  3,  3,  4,  4,  5,  5,  6,  6, 7, 7, 8, 8, ...
  1, 2, 3,  5,  6,  9, 10, 14, 15, 20, 21, 27, ...
  1, 2, 5,  7, 13, 16, 26, 30, 45, ...
  1, 3, 6, 13, 19, 35, 45, 75, ...
  1, 3, 9, 16, 35, 51, 96, ...
  ...
From _Peter Bala_, Aug 19 2021: (Start)
With the arrays M(k) as defined in the Comments section, the infinite product M(0)*M(1)*M(2)*... begins
  /1        \/1        \/1       \ /1       \        /1         \
  |1 1      ||0 1      ||0 1      ||0 1      |       |1 1       |
  |1 0 1    ||0 1 1    ||0 0 1    ||0 0 1    |...  = |1 1 1     |
  |1 0 1 1  ||0 1 0 1  ||0 0 1 1  ||0 0 0 1  |       |1 2 2 1   |
  |1 0 1 0 1||0 1 0 1 1||0 0 1 0 1||0 0 0 1 1|       |1 2 3 2 1 |
  |...      ||...       |...      ||...      |       |...       |
(End)

Rémy Sigrist, Rows 0..199, flattened
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Richard Ehrenborg and Margaret A. Readdy, The Gaussian coefficient revisited, arXiv:1609.03216 [math.CO], 2016.
Richard L. Ollerton and Anthony G. Shannon, Some properties of generalized Pascal squares and triangles, Fib. Q., 36 (No. 2, 1998), 98-109. See Table 10.
Wikipedia, Hockey-stick identity.

A123149 is essentially the same triangle, except for a diagonal of zeros.
Row sums are in A182522 (essentially A038754).
Cf. A027907, A070909.
See A295555 for the next triangle in the series A007318, A169623 (this sequence).

T:=proc(n,k) option remember;
if n >= 0 and k = 0 then 1
elif n >= 0 and k = n then 1
elif (k < 0 or k > n) then 0
else T(n-2,k-2)+T(n-2,k-1)+T(n-2,k);
fi;
end;
for n from 0 to 14 do lprint([seq(T(n,k),k=0..n)]); od: # N. J. A. Sloane, Nov 23 2017

p[x, 1] := 1;
p[x_, n_] := p[x, n] = If[Mod[n, 2] == 0, (x + 1)*p[x, n - 1], (x^2 + x + 1)^Floor[n/2]]
a = Table[CoefficientList[p[x, n], x], {n, 1, 12}]
Flatten[a] (* This is for the same sequence but with offset 1 *)

From Peter Bala, Aug 19 2021: (Start)
T(2*n,k) = T(2*n-1,k-1) + T(2*n-2,k).
T(2*n,k) = T(2*n-1,k) + T(2*n-2,k-2).
T(2*n+1,k) = T(2*n,k) + T(2*n,k-1).
Hockey stick identities (relate row k entries to entries in row k-1):
T(2*n,k) = T(2*n-1,k-1) + T(2*n-3,k-1) + T(2*n-5,k-1) + ....
T(2*n+1,k) = T(2*n,k-1) + ( T(2*n-1,k-1) + T(2*n-3,k-1) + T(2*n-5,k-1) + ... ). (End)

Keyword:tabl added, notation standardized, formula added by the Assoc. Editors of the OEIS, Feb 02 2010

Entry revised by N. J. A. Sloane, Nov 23 2017

cp's OEIS Frontend

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A177991 Triangle read by rows, A070909 * A177990.

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A038754 a(2n) = 3^n, a(2n+1) = 2*3^n.

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