A128174 -id:A128174 - cp's OEIS frontend

A134511 abs(A049310) * A128174 provided both arrays are read with offset (n,k) = (0,0).

1, 0, 1, 2, 0, 1, 0, 3, 0, 1, 5, 0, 4, 0, 1, 0, 8, 0, 5, 0, 1, 13, 0, 12, 0, 6, 0, 1, 0, 21, 0, 17, 0, 7, 0, 1, 34, 0, 33, 0, 23, 0, 8, 0, 1, 0, 55, 0, 50, 0, 30, 0, 9, 0, 1, 89, 0, 88, 0, 73, 0, 38, 0, 10, 0, 1, 0, 144, 0, 138, 0, 103, 0, 47, 0, 11, 0, 1, 233, 0, 232, 0, 211, 0, 141, 0, 57, 0, 12, 0, 1

Offset: 0

Gary W. Adamson, Oct 28 2007

A112552(unsigned) = A128174 * A049310.
Row sums = A134512: (1, 1, 3, 4, 10, 14, 32, 46, 99, 145, ...).
From Petros Hadjicostas, Sep 03 2019: (Start)
To prove Alois P. Heinz's claim (see the Formula section and his Maple program below) we note that, for n >= 0 and 0 <= k <= n, T(n, n-k) = Sum_{r = 0 .. infinity} abs(A049310(n,r)) * A128174(r,n-k) = Sum_{r = n-k..n} abs(A049310(n,r)) * A128174(r,n-k). But A049310(n,r) = 0 when n + r is odd and A128174(r,n-k) = 1 iff r + n - k is even. Thus, when k is odd, T(n, n-k) = 0.
Assume now k is even. Then T(n, n-k) = Sum_{r = n-k..n and n+r even} abs(A049310(n,r)) =  Sum_{r = n-k..n and n+r even} binomial((n+r)/2, r). Letting m = n-r (which is even), we see that the summation ranges from m = 0 to k over even numbers. Thus, let s = m/2, and so T(n, n-k) = Sum_{s = 0 .. k/2} binomial(n-s, n-2*s) = Sum_{s = 0 .. k/2} binomial(n-s, s) = F(n+1, k/2), where F(.,.) is the incomplete Fibonacci number from the references (see also the Formula section below).
(End)

			First few rows of the triangle T(n,k):
   1;
   0,  1;
   2,  0,  1;
   0,  3,  0,  1;
   5,  0,  4,  0,  1;
   0,  8,  0,  5,  0,  1;
  13,  0, 12,  0,  6,  0,  1;
   0, 21,  0, 17,  0,  7,  0,  1;
  34,  0, 33,  0, 23,  0,  8,  0,  1;
   0, 55,  0, 50,  0, 30,  0,  9,  0,  1;
  ...

Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 141, flattened)
H. Belbachir and A. Belkhir, Combinatorial Expressions Involving Fibonacci, Hyperfibonacci, and Incomplete Fibonacci Numbers, Journal of Integer Sequences, Vol. 17 (2014), Article 14.4.3.
A. Dil and I. Mezo, A symmetric algorithm for hyperharmonic and Fibonacci numbers, Appl. Math. Comp. 206 (2008), 942-951; in Eqs. (11), see the incomplete Fibonacci numbers.
Piero Filipponi, Incomplete Fibonacci and Lucas numbers, P. Rend. Circ. Mat. Palermo (Serie II) 45(1) (1996), 37-56; see Table 1 (p. 39) that contains the incomplete Fibonacci numbers.
A. Pintér and H.M. Srivastava, Generating functions of the incomplete Fibonacci and Lucas numbers, Rend. Circ. Mat. Palermo (Serie II) 48(3) (1999), 591-596.

Cf. A038730, A038792, A049310, A128174, A134512.

A(4n,2n) gives: A038736.

N:= 20: # for the first N rows
T128174:= Matrix(N,N,(i,j) -> `if`(j<=i, (i-j+1) mod 2, 0)):
T049310:= Matrix(N,N):
for i from 1 to N do
     P:= orthopoly[U](i-1,x/2);
     for j from 1 to i do
       T049310[i,j]:= abs(coeff(P,x,j-1))
     od
od:
A:= T049310 . T128174:
for i from 1 to N do
convert(A[i,1..i],list)
od;  # Robert Israel, Mar 02 2018
# second Maple program:
T:= (n, k)-> `if`((n+k)::odd, 0, add(binomial(n-s, s), s=0..(n-k)/2)):
seq(seq(T(n, k), k=0..n), n=0..12); # Alois P. Heinz, Sep 02 2019

T[n_, k_] := If[OddQ[n+k], 0, Sum[Binomial[n-s, s], {s, 0, (n-k)/2}]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 31 2021, after Alois P. Heinz *)

abs(A049310) * A128174 as infinite lower triangular matrices assuming both of them have offset (n,k) = (0,0).
From Petros Hadjicostas, Sep 03 2019: (Start)
Let F(m,r) = Sum_{j = 0..r} binomial(m-1-j, j) be the incomplete Fibonacci numbers from the references (defined for m >= 1 and 0 <= r <= floor((m-1)/2)).
As Alois P. Heinz observed, for n >= 0 and 0 <= k <= n, T(n, n-k) = F(n+1, k/2) when k is even, and = 0 otherwise (see his Maple program below).
(End)

Edited by Robert Israel, Mar 02 2018

A128175 Binomial transform of A128174.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 4, 3, 1, 8, 8, 7, 4, 1, 16, 16, 15, 11, 5, 1, 32, 32, 31, 26, 16, 6, 1, 64, 64, 63, 57, 42, 22, 7, 1, 128, 128, 127, 120, 99, 64, 29, 8, 1, 256, 256, 255, 247, 219, 163, 93, 37, 9, 1

Offset: 1

Gary W. Adamson, Feb 17 2007

Row sums = A045623: (1, 2, 5, 12, 28, 64, 144, ...).
A128176 = A128174 * A007318.
Riordan array ((1-x)/(1-2x),x/(1-x)). - Paul Barry, Oct 02 2010
Fusion of polynomial sequences p(n,x) = (x+1)^n and q(n,x) = x^n + x^(n-1) + ... + x + 1; see A193722 for the definition of fusion. - Clark Kimberling, Aug 04 2011

			First few rows of the triangle:
   1;
   1,  1;
   2,  2,  1;
   4,  4,  3,  1;
   8,  8,  7,  4,  1;
  16, 16, 15, 11,  5,  1;
  32, 32, 31, 26, 16,  6,  1;
  64, 64, 63, 57, 42, 22,  7,  1;
  ...
From _Paul Barry_, Oct 02 2010: (Start)
Production matrix is
  1, 1;
  1, 1, 1;
  0, 0, 1, 1;
  0, 0, 0, 1, 1;
  0, 0, 0, 0, 1, 1;
  0, 0, 0, 0, 0, 1, 1;
  0, 0, 0, 0, 0, 0, 1, 1;
  0, 0, 0, 0, 0, 0, 0, 1, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 1, 1;
  ...
Matrix logarithm is
  0;
  1, 0;
  1, 2, 0;
  1, 1, 3, 0;
  1, 1, 1, 4, 0;
  1, 1, 1, 1, 5, 0;
  1, 1, 1, 1, 1, 6, 0;
  1, 1, 1, 1, 1, 1, 7, 0;
  1, 1, 1, 1, 1, 1, 1, 8, 0;
  1, 1, 1, 1, 1, 1, 1, 1, 9,  0;
  1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 0;
  ... (End)
.
First few rows of the array:
  1, 1,  2,  4,  8,  16, ...
  1, 2,  4,  8, 16,  32, ...
  1, 3,  7, 15, 31,  63, ...
  1, 4, 11, 26, 57, 120, ...
  1, 5, 16, 42, 99, 219, ...
  ...

Cf. A045623, A128176, A007318.

A193820 := (n,k) -> `if`(k=0 or n=0, 1, A193820(n-1,k-1)+A193820(n-1,k));
A128175 := (n,k) -> A193820(n-1,n-k);
seq(print(seq(A128175(n,k),k=0..n)),n=0..10); # Peter Luschny, Jan 22 2012

z = 10; a = 1; b = 1;
p[n_, x_] := (a*x + b)^n
q[0, x_] := 1
q[n_, x_] := x*q[n - 1, x] + 1; q[n_, 0] := q[n, x] /. x -> 0;
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]   (* A193820 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A128175 *)
(* Clark Kimberling, Aug 06 2011 *)
(* function dotTriangle[] is defined in A128176 *)
a128175[r_] := dotTriangle[Binomial, If[EvenQ[#1 + #2], 1, 0]&, r]
TableForm[a128174[7]] (* triangle *)
Flatten[a128174[9]] (* data *) (* Hartmut F. W. Hoft, Mar 15 2017 *)

A007318 * A128174 as infinite lower triangular matrices.
Antidiagonals of an array in which the first row = (1, 1, 2, 4, 8, 16, ...); and (n+1)-th row = partial sums of n-th row.
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(4 + 4*x + 3*x^2/2! + x^3/3!) = 4 + 8*x + 15*x^2/2! + 26*x^3/3! + 42*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 22 2014
T(n, k) = Sum_{i=0..floor((n-k)/2)} binomial(n-1, k-1+2*i). - Werner Schulte, Mar 05 2025
T(n, k) = binomial(n-1, k-1)*hypergeom([1, (k-n)/2, (1+k-n)/2], [(1+k)/2, k/2], 1). - Stefano Spezia, Mar 07 2025

A130125 Triangle defined by A128174 * A130123, read by rows.

Original entry on oeis.org

1, 0, 2, 1, 0, 4, 0, 2, 0, 8, 1, 0, 4, 0, 16, 0, 2, 0, 8, 0, 32, 1, 0, 4, 0, 16, 0, 64, 0, 2, 0, 8, 0, 32, 0, 128, 1, 0, 4, 0, 16, 0, 64, 0, 256, 0, 2, 0, 8, 0, 32, 0, 128, 0, 512, 1, 0, 4, 0, 16, 0, 64, 0, 256, 0, 1024, 0, 2, 0, 8, 0, 32, 0, 128, 0, 512, 0, 2048

Offset: 0

Gary W. Adamson, May 11 2007

Row sums = A000975: (1, 2, 5, 10, 21, 42, ...).

			First few rows of the triangle are:
  1;
  0, 2;
  1, 0, 4;
  0, 2, 0, 8;
  1, 0, 4, 0, 16;
  0, 2, 0, 8,  0, 32; ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A000975, A128174, A130123.

Flat(List([0..10], n-> List([0..n], k-> 2^(k-1)*(1+(-1)^(n-k)) ))); # G. C. Greubel, Jun 05 2019

[[2^(k-1)*(1+(-1)^(n-k)): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Jun 05 2019

Table[2^(k-1)*(1+(-1)^(n-k)), {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 05 2019 *)

{T(n,k) = 2^(k-1)*(1+(-1)^(n-k))}; \\ G. C. Greubel, Jun 05 2019

[[2^(k-1)*(1+(-1)^(n-k)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jun 05 2019

A128174 * A130123 as infinite lower triangular matrices. By columns, (2^k, 0, 2^k, 0, ...).

T(n,k) = 2^(k-1)*(1 + (-1)^(n-k)). - G. C. Greubel, Jun 05 2019

More terms added by G. C. Greubel, Jun 05 2019

A128176 A128174 * A007318.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 4, 3, 1, 3, 6, 7, 4, 1, 3, 9, 13, 11, 5, 1, 4, 12, 22, 24, 16, 6, 1, 4, 16, 34, 46, 40, 22, 7, 1, 5, 20, 50, 80, 86, 62, 29, 8, 1, 5, 25, 70, 130, 166, 148, 91, 37, 9, 1, 6, 30, 95, 200, 296, 314, 239, 128, 46, 10, 1

Offset: 1

Gary W. Adamson, Feb 17 2007

Row Sums = A000975: (1, 2, 5, 10, 21, 42, 85, 170, ...).
A007318 * A128174 = A128175.
From Peter Bala, Aug 14 2014: (Start)
Riordan array ( 1/((1 - x^2)*(1 - x)), x/(1 - x) ).
Let B_n be the set of length n nonzero binary words ending in an even number (possibly 0) of 0's. Then T(n,k) is the number of words in B_n having k 1's. An example is given below. (End)

			First few rows of the triangle are:
  1;
  1,  1;
  2,  2,  1;
  2,  4,  3,  1;
  3,  6,  7,  4,  1;
  3,  9, 13, 11,  5,  1;
  4, 12, 22, 24, 16,  6,  1;
  4, 16, 34, 46, 40, 22,  7,  1;
  ...
From _Peter Bala_, Aug 14 2014: (Start)
Row 4: [2,4,3,1].
k      Binary words in B_4 with k 1's       Number
- - - - - - - - - - - - - - - - - - - - - - - - - -
1      0001, 0100                            2
2      0011, 0101, 1001, 1100                4
3      0111, 1011, 1101                      3
4      1111                                  1
- - - - - - - - - - - - - - - - - - - - - - - - - -
The infinitesimal generator matrix begins
   0
   1  0
   1  2  0
  -1  1  3  0
   1 -1  1  4  0
  -1  1 -1  1  5  0
  ...
Cf. A132440. (End)

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, Part IV, 4. Mitteilungen zur Lehre vom Transfiniten, VIII Nr. 13, Springer, Berlin, 1932. See p. 434.

Cf. A000975, A128175, A007318.

Cf. A035317 (mirror). [Johannes W. Meijer, Jul 20 2011]

(* Dot product of two lower triangular matrices *)
dotRow[r_, s_, n_] := Map[Sum[r[n, k] s[k, #], {k, #, n}]&, Range[0, n]]
dotTriangle[r_, s_, n_] := Map[dotRow[r, s, #]&, Range[0, n]]
(* The pure function in the first argument computes A128174 *)
a128176[r_] := dotTriangle[If[EvenQ[#1 + #2], 1, 0]&, Binomial, r]
TableForm[a128176[7]] (* triangle *)
Flatten[a128176[9]] (* data *) (* Hartmut F. W. Hoft, Mar 15 2017 *)
T[n_, n_] := 1; T[n_, 0] := 1 + Floor[n/2]; T[n_, k_] := T[n, k] = T[n - 1, k - 1] + T[n - 1, k]; Table[T[n, k], {n,0,20}, {k, 0, n}] // Flatten (* G. C. Greubel, Sep 30 2017 *)

for(n=0, 10, for(k=0,n, print1(sum(i=0,floor(n/2), binomial(n - 2*i,k)), ", "))) \\ G. C. Greubel, Sep 30 2017

A128174 * A007318 (Pascal's triangle), as infinite lower triangular matrices.
From Peter Bala, Aug 14 2014: (Start)
Working with a row and column offset of 0 we have T(n,k) = Sum_{i = 0..floor(n/2)} binomial(n - 2*i,k).
O.g.f.: 1/( (1 - z^2)*(1 - z*(1 + x)) ) = Sum_{n >= 0} R(n,x)*z^n = 1 + (1 + x)*z + (2 + 2*x + x^2)*z^2 + ....
The row polynomials satisfy R(n+2,x) - R(n,x) = (1 + x)^(n+1). (End)
From Hartmut F. W. Hoft, Mar 15 2017: (Start)
Using offset 0, the triangle has the Pascal Triangle recursion pattern:
T(n, 0) = 1 + floor(n/2) and T(n, n) = 1, for n >= 0;
T(n, k) = T(n-1, k-1) + T(n-1, k) for n > 0 and 0 < k < n. (End)

A128187 Matrix product A128174 * A051731 read by rows.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Mar 07 2007

			First few rows of the triangle are:
1;
1, 1;
2, 0, 1;
2, 2, 0, 1;
3, 0, 1, 0, 1;
3, 3, 1, 1, 0, 1;
4, 0, 1, 0, 1, 0, 1;
4, 4, 1, 2, 0, 1, 0, 1;
...

Cf. A128188 (row sums), A128174, A051731.

A128187 := proc(n,k)
    add( A128174(n,j)*A051731(j,k), j=k..n) ;
end proc: # R. J. Mathar, Apr 16 2013

A128174 * A051731 as infinite lower triangular matrices.

A128222 A127701 * A128174.

Original entry on oeis.org

1, 1, 2, 3, 1, 3, 1, 4, 1, 4, 5, 1, 5, 1, 5, 1, 6, 1, 6, 1, 6, 7, 1, 7, 1, 7, 1, 7, 1, 8, 1, 8, 1, 8, 1, 8, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10

Offset: 1

Gary W. Adamson, Feb 19 2007

Row sums = A128223: (1, 3, 7, 10, 17, 21, 31, 36, ...).

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

Cf. A127701, A128174, A128223, A128222.

a128222[n_, k_] := If[EvenQ[n-k], n, 1]/;1<=k<=n
a128222[r_] := Table[a128222[n, k], {n, 1, r}, {k, 1, n}]
TableForm[a128222[7]] (* triangle *)
Flatten[a128222[10]] (* data *) (* Hartmut F. W. Hoft, Mar 08 2017 *)

A127701 * A128174 as infinite lower triangular matrices. Odd rows: n terms of n, 1, n, ...; even rows: n terms of 1, n, 1, ...

Inserted omitted values a(28) = 7 and a(29) = 1, Hartmut F. W. Hoft, Mar 08 2017

A128619 Triangle T(n, k) = A127647(n,k) * A128174(n,k), read by rows.

Original entry on oeis.org

1, 0, 1, 2, 0, 2, 0, 3, 0, 3, 5, 0, 5, 0, 5, 0, 8, 0, 8, 0, 8, 13, 0, 13, 0, 13, 0, 13, 0, 21, 0, 21, 0, 21, 0, 21, 34, 0, 34, 0, 34, 0, 34, 0, 34, 0, 55, 0, 55, 0, 55, 0, 55, 0, 55

Offset: 1

Gary W. Adamson, Mar 14 2007

This triangle is different from A128618, which is equal to A128174 * A127647.

			First few rows of the triangle are:
   1;
   0,  1;
   2,  0,  2;
   0,  3,  0,  3;
   5,  0,  5,  0,  5;
   0,  8,  0,  8,  0,  8;
  13,  0, 13,  0, 13,  0, 13;
   0, 21,  0, 21,  0, 21,  0, 21,
  ...

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A000045, A096140, A127646, A128174, A128610, A128618, A128620.

Cf. A128620 (row sums).

[((n+k+1) mod 2)*Fibonacci(n): k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 17 2024

Table[Fibonacci[n]*Mod[n+k+1,2], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 16 2024 *)

flatten([[((n+k+1)%2)*fibonacci(n) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 17 2024

T(n, k) = A127647 * A128174, an infinite lower triangular matrix. In odd rows, n terms of F(n), 0, F(n),...; in the n-th row. In even rows, n terms of 0, F(n), 0,...; in the n-th row.
Sum_{k=1..n} T(n, k) = A128620(n-1).
From G. C. Greubel, Mar 16 2024: (Start)
T(n, k) = Fibonacci(n)*(1 + (-1)^(n+k))/2.
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^n*A128620(n-1).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/2)*(1-(-1)^n)*A096140(floor((n + 1)/2)).
Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = (1/2)*(1 - (-1)^n)*( Fibonacci(n-1) + (-1)^floor((n-1)/2) * Fibonacci(floor((n-3)/2)) ). (End)

A128621 A127648 * A128174 as an infinite lower triangular matrix.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Mar 14 2007

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

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A000326, A123684, A127648, A128174.

Cf. A093005 (row sums).

[n*(1+(-1)^(n+k))/2: k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 13 2024

Table[n*(1+(-1)^(n+k))/2, {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 13 2024 *)

flatten([[n*(1+(-1)^(n+k))//2 for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 13 2024

Odd rows: n terms of n, 0, n, ...; even rows, n terms of 0, n, 0, ...
T(n,k) = n if n+k even, T(n,k) = 0 if n+k odd.
Sum_{k=1..n} T(n, k) = A093005(n) (row sums).
From G. C. Greubel, Mar 13 2024: (Start)
T(n, k) = n*(1 + (-1)^(n+k))/2.
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n+1)*A093005(n).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/2)*(1-(-1)^n) * A000326(floor((n+1)/2)).
Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = (1/2)*(1 - (-1)^n)*A123684(floor((n+1)/2)). (End)

More terms added by G. C. Greubel, Mar 13 2024

A131406 3A128174 - 2A000012(signed).

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1

Offset: 1

Gary W. Adamson, Jul 07 2007

Row sums = A032766, congruent to {0,1} mod 3: (1, 3, 4, 6, 7, 9, 10,...).

Sequence array for the expansion of (1+2x)/(1-x^2). A105476 is an eigensequence. [From Paul Barry, Nov 03 2010]

			First few rows of the triangle are:
1;
2, 1;
1, 2, 1;
2, 1, 2, 1;
1, 2, 1, 2, 1;
2, 1, 2, 1, 2, 1;
...

Cf. A128174, A000012, A032766.

T[n_, k_] := Mod[n-k, 2]+1; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 17 2016 *)

3*A128174 - 2*A000012(signed + - + 1 by columns). (1, 2, 1, 2, 1,...) in every column.

Triangle T(n,k)=if(k<=n,(3-(-1)^(n-k))/2). [From Paul Barry, Nov 03 2010]

A135839 Triangle read by rows: starting with A128174, replace left border with (1, 1, 1, ...).

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Dec 01 2007

Row sums = (1, 2, 2, 3, 3, 4, 4, 5, 5, ...).

			First few rows of the triangle are:
  1;
  1, 1;
  1, 0, 1;
  1, 1, 0, 1;
  1, 0, 1, 0, 1;
  1, 1, 0, 1, 0, 1;
  1, 0, 1, 0, 1, 0, 1;
  ...

Zhuorui He, Rows 1..150 of the triangle, flattened (first 50 rows from G. C. Greubel)

T[1, 1] := 1; T[n_, 1] := 1; T[n_, n_] := 1; T[n_, k_] := (1 - (-1)^(n - k + 1))/2; Table[T[n, k], {n, 1, 10}, {k, 1, n}]//Flatten (* G. C. Greubel, Dec 05 2016 *)
Flatten[Table[Join[{1},PadLeft[{},n,{0,1}]],{n,0,20}]] (* Harvey P. Dale, Feb 26 2024 *)

Given A128174, replace left border with (1, 1, 1, ...). Triangle read by rows, odd rows = n terms of (1, 0, 1, ...); even rows = n terms of (1, 1, 0, 1, 0, 1, ...). By columns, leftmost column = (1, 1, 1, ...); all others = (1, 0, 1, 0, 1, ...).

T(n,1) = T(n,n) = 1, T(n,k) = (1 - (-1)^(n-k-1))/2. - G. C. Greubel,Dec 05 2016

Definition corrected by Zhuorui He, Jul 21 2025

cp's OEIS Frontend

A134511 abs(A049310) * A128174 provided both arrays are read with offset (n,k) = (0,0).

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A128175 Binomial transform of A128174.

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A130125 Triangle defined by A128174 * A130123, read by rows.

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A128176 A128174 * A007318.

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A128187 Matrix product A128174 * A051731 read by rows.

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A128222 A127701 * A128174.

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A128619 Triangle T(n, k) = A127647(n,k) * A128174(n,k), read by rows.

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A131406 3A128174 - 2A000012(signed).