A128174 -id:A128174 - cp's OEIS frontend

A136579 Triangle read by rows: A128174 * A136572.

1, 0, 1, 1, 0, 2, 0, 1, 0, 6, 1, 0, 2, 0, 24, 0, 1, 0, 6, 0, 120, 1, 0, 2, 0, 24, 0, 720, 0, 1, 0, 6, 0, 120, 0, 5040, 1, 0, 2, 0, 24, 0, 720, 0, 40320

Offset: 0

Gary W. Adamson, Jan 09 2008

Row sums = A136580: 1, 1, 3, 7, 27, 127, ...

			First few rows of the triangle:
  1;
  0, 1;
  1, 0, 2;
  0, 1, 0, 6;
  1, 0, 2, 0, 24;
  0, 1, 0, 6,  0, 120;
  1, 0, 2, 0, 24,   0, 720;
  ...

Cf. A136580, A128174, A136572, A000142.

A128174 * A136572 Triangle, even rows = even n! interspersed with zeros. Odd n rows, = odd n! interspersed with zeros.

T(2*i,2*k) = (2*k)! = A010050(k). T(2*i+1,2*k+1) = (2*k+1)! = A009445(k). - R. J. Mathar, Jun 04 2021

A128221 A128174 * A127701.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Feb 19 2007

Row sums = A024206: (1, 3, 5, 8, 11, 15, 19, ...). A128222 = A127701 * A128174.

Table T(n,k) = n, if k is odd, 1 if k is even; n, k > 0, read by antidiagonals. -Boris Putievskiy, Jan 30 2013

			From _Boris Putievskiy_, Jan 30 2013: (Start)
The start of the sequence as a table:
  1, 1, 1, 1, 1, 1, 1, ...
  2, 1, 2, 1, 2, 1, 2, ...
  3, 1, 3, 1, 3, 1, 3, ...
  4, 1, 4, 1, 4, 1, 4, ...
  5, 1, 5, 1, 5, 1, 5, ...
  6, 1, 6, 1, 6, 1, 6, ...
  7, 1, 7, 1, 7, 1, 7, ...
  ...
(End)
First few rows of the triangle are:
  1;
  1, 2;
  1, 1, 3;
  1, 2, 1, 4;
  1, 1, 3, 1, 5;
  1, 2, 1, 4, 1, 6;
  1, 1, 3, 1, 5, 1, 7;
  ...

Robert G. Wilson v, Table of n, a(n) for n = 1..1000

Cf. A128174, A127701, A024206, A128222.

a128221[n_, k_] := If[EvenQ[n-k], k, 1]/;1<=k<=n
a128221[r_] := Table[a128221[n, k], {n, 1, r}, {k, 1, n}]
TableForm[a128221[7]] (* triangle *)
Flatten[a128221[10]] (* data *) (* Hartmut F. W. Hoft, Mar 08 2017 *)
t[r_, c_] := If[ OddQ@ c, r, 1]; Table[t[k, n - k + 1], {n, 13}, {k, n}] // Flatten (* Robert G. Wilson v, Mar 09 2017 *)

A128174 * A127701 as infinite lower triangular matrices. By columns, k-th column = k, 1, k, ...; k=1,2,3,...
From Boris Putievskiy, Jan 30 2013: (Start)
As table T(n,k) = (1+(-1)^k)/2 - (-1+(-1)^k)*n/2.
As linear sequence a(n) = (1+(-1)^A004736(n))/2 - (-1+(-1)^A004736(n))*A002260(n)/2. a(n) = (1+(-1)^j)/2 - (-1+(-1)^j)*i/2,
where i = n-t*(t+1)/2, j = (t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). (End)

More terms from Robert G. Wilson v, Mar 09 2017

A128618 Triangle read by rows: A128174 * A127647 as infinite lower triangular matrices.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 0, 1, 0, 3, 1, 0, 2, 0, 5, 0, 1, 0, 3, 0, 8, 1, 0, 2, 0, 5, 0, 13, 0, 1, 0, 3, 0, 8, 0, 21, 1, 0, 2, 0, 5, 0, 13, 0, 34, 0, 1, 0, 3, 0, 8, 0, 21, 0, 55, 1, 0, 2, 0, 5, 0, 13, 0, 34, 0, 89, 0, 1, 0, 3, 0, 8, 0, 21, 0, 55, 0, 144, 1, 0, 2, 0, 5, 0, 13, 0, 34, 0, 89, 0, 233

Offset: 1

Gary W. Adamson, Mar 14 2007

This triangle is different from A128619, which is A128619 = A127647 * A128174.

			First few rows of the triangle are:
  1;
  0, 1;
  1, 0, 2;
  0, 1, 0, 3;
  1, 0, 2, 0, 5;
  0, 1, 0, 3, 0, 8;
  1, 0, 2, 0, 5, 0, 13;
  0, 1, 0, 3, 0, 8,  0, 21;
  1, 0, 2, 0, 5, 0, 13,  0, 34;
  0, 1, 0, 3, 0, 8,  0, 21,  0, 55;
  1, 0, 2, 0, 5, 0, 13,  0, 34,  0, 89;
  ...

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

Cf. A000045, A052952, A127647, A128174, A128619, A355020.

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

Table[Fibonacci[k]*Mod[n-k+1,2], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 17 2024 *)

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

By columns, Fibonacci(k) interspersed with alternate zeros in every column, k=1,2,3,...
Sum_{k=1..n} T(n, k) = A052952(n-1) (row sums).
From G. C. Greubel, Mar 17 2024: (Start)
T(n, k) = (1/2)*(1 + (-1)^(n+k))*Fibonacci(k).
T(n, n) = A000045(n).
T(2*n-1, n) = (1/2)*(1-(-1)^n)*A000045(n).
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n-1)*A052952(n-1).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/2)*(1 - (-1)^n)*(Fibonacci((n+ 5)/2) - 1).
Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = (1/2)*(1-(-1)^n) *  A355020(floor((n-1)/2)). (End)

a(6) corrected and more terms from Georg Fischer, May 30 2023

A128622 Triangle T(n, k) = A128064(unsigned) * A128174, read by rows.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Mar 14 2007

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

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

Cf. A000027, A016777, A042963, A047461, A109613, A128064, A128174.

Cf. A000326 (diagonal sums), A014848 (row sums), A319556 (alternating row sums).

[n - ((n+k) mod 2): k in [1..n], n in [1..16]]; // G. C. Greubel, Mar 14 2024

Table[n - Mod[n+k,2], {n,16}, {k,n}]//Flatten (* G. C. Greubel, Mar 14 2024 *)

flatten([[n - ((n+k)%2) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 14 2024

T(n, k) = abs(A128064(n,k) * A128174(n, k), as infinite lower triangular matrices.
Sum_{k=1..n} T(n, k) = A014848(n) (row sums).
From G. C. Greubel, Mar 14 2024: (Start)
T(n, k) = n - (1 - (-1)^(n+k))/2 = n - (n+k mod 2).
T(n, 1) = A109613(n+1).
T(n, n) = A000027(n).
T(2*n-1, n) = A042963(n).
T(3*n-1, n) = A016777(n+1).
T(4*n-3, n) = A047461(n).
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A319556(n).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A000326(floor((n+1)/2)).
Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = A123684(floor((n+1)/2)). (End)

More terms added by G. C. Greubel, Mar 14 2024

A129688 A129686 * A128174.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Apr 28 2007

Resulting triangle has (1, 0, 2, 0, 2, 0, 2, ...) in every column.

Row sums = A109613: (1, 1, 3, 3, 5, 5, ...).

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

Cf. A129686, A128174, A109613.

T[n_, k_] := If[k == n, 1, If[EvenQ[n + k], 2, 0]];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 10 2019 *)

A129686 * A128174 as infinite lower triangular matrices.

A131228 3A051340 - 2A128174.

Original entry on oeis.org

1, 3, 4, 1, 3, 7, 3, 1, 3, 10, 1, 3, 1, 3, 13, 3, 1, 3, 1, 3, 16, 1, 3, 1, 3, 1, 3, 19, 3, 1, 3, 1, 3, 1, 3, 22, 1, 3, 1, 3, 1, 3, 25, 3, 1, 3, 1, 3, 1, 3, 1, 3, 28

Offset: 0

Gary W. Adamson, Jun 20 2007

nonn

Row sums = A131229.

			First few rows of the triangle are:
1;
3, 4;
1, 3, 7;
3, 1, 3, 10;
1, 3, 1, 3, 13;
...

Cf. A051340, A128174, A131229, A131227.

3*A051340 - 2*A128174 as infinite lower triangular matrices.

A133728 A128174 * A127775.

Original entry on oeis.org

1, 0, 3, 1, 0, 5, 0, 3, 0, 7, 1, 0, 5, 0, 9, 0, 3, 0, 7, 0, 11, 1, 0, 5, 0, 9, 0, 13, 0, 3, 0, 7, 0, 11, 0, 15, 1, 0, 5, 0, 9, 0, 13, 0, 17, 0, 3, 0, 7, 0, 11, 0, 15, 0, 19, 1, 0, 5, 0, 9, 0, 13, 0, 17, 0, 21, 0, 3, 0, 7, 0, 11, 0, 15, 0, 19, 0, 23

Offset: 1

Gary W. Adamson, Sep 21 2007

Row sums are the triangular numbers 1, 3, 6, 10, 15, 21, 28, ...; see A000217.

			From _Philippe Deléham_, Oct 28 2011: (Start)
Triangle begins:
  1;
  0,  3;
  1,  0,  5;
  0,  3,  0,  7;
  1,  0,  5,  0,  9;
  0,  3,  0,  7,  0, 11;
  1,  0,  5,  0,  9,  0, 13; ...
(End)

Cf. A000217.

A128174 * A127775 as infinite lower triangular matrices. Triangle by columns: (2k-1, 0, 2k-1, 0, ...) in k-th column.

Corrected and extended by Philippe Deléham, Oct 28 2011

A134317 Triangle, A128174 * A134309 as infinite lower triangular matrices.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 0, 1, 0, 4, 1, 0, 2, 0, 8, 0, 1, 0, 4, 0, 16, 1, 0, 2, 0, 8, 0, 32, 0, 1, 0, 4, 0, 16, 0, 64, 1, 0, 2, 0, 8, 0, 32, 0, 128, 0, 1, 0, 4, 0, 16, 0, 64, 0, 256

Offset: 1

Gary W. Adamson, Oct 19 2007

Is this the same as A123641? - R. J. Mathar, Mar 28 2012

			First few rows of the triangle are:
1;
0, 1;
1, 0, 2;
0, 1, 0, 4;
1, 0, 2, 0, 8;
0, 1, 0, 4, 0, 16;
1, 0, 2, 0, 8, 0, 32;
0, 1, 0, 4, 0, 16, 0, 64;
...

Cf. A128174, A134309, A001045 (row sums).

T(n,k) = 0 if n+k odd, else T(n,1) =1, T(n,k)=2^(k-2) if k>=2. - R. J. Mathar, Sep 01 2024

A134352 A130123 * A128174.

Original entry on oeis.org

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

Offset: 0

Gary W. Adamson, Oct 21 2007

Row sums = A134353.

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

Cf. A130123, A128174, A134353.

A130123 * A128174 as infinite lower triangular matrices.

Triangle read by rows: even n-th row = n+1 terms of (2^n, 0, 2^n, ...); odd n-th row = n+1 terms of (0, 2^n, 0, 2^n, ...).

A134446 A128174 * A002260.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 2, 4, 3, 4, 3, 4, 6, 4, 5, 3, 6, 6, 8, 5, 6, 4, 6, 9, 8, 10, 6, 7, 4, 8, 9, 12, 10, 12, 7, 8, 5, 8, 12, 12, 15, 12, 14, 8, 9, 5, 10, 12, 16, 15, 18, 14, 16, 9, 10

Offset: 1

Gary W. Adamson, Oct 25 2007

Row sums = A002623: (1, 3, 7, 13, 22, 34, ...).

			First few rows of the triangle:
  1;
  1,  2;
  2,  2,  3;
  2,  4,  3,  4;
  3,  4,  6,  4,  5;
  3,  6,  6,  8,  5,  6;
  4,  6,  9,  8, 10,  6,  7;
  ...

Cf. A002260, A128174, A002623.

A128174 * A002260 as infinite lower triangular matrices.

cp's OEIS Frontend

A136579 Triangle read by rows: A128174 * A136572.

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A128221 A128174 * A127701.

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A128618 Triangle read by rows: A128174 * A127647 as infinite lower triangular matrices.

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A128622 Triangle T(n, k) = A128064(unsigned) * A128174, read by rows.

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A129688 A129686 * A128174.

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A131228 3*A051340 - 2*A128174.

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A133728 A128174 * A127775.

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A134317 Triangle, A128174 * A134309 as infinite lower triangular matrices.

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A131228 3A051340 - 2A128174.