A090172 -id:A090172 - cp's OEIS frontend

A090171 Triangle read by rows, related to Pascal's triangle read mod 2, starting with 0, 1, 0.

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

Offset: 0

N. J. A. Sloane, Jan 19 2004

			Triangle begins:
0
1 0
1 1 0
0 1 0 0
1 1 1 1 0
...

Y. Moshe, The density of 0's in recurrence double sequences, J. Number Theory, 103 (2003), 109-121; see Fig. 2.

Cf. A007318, A083093.
Cf. A090172, A090173, A090174, A091533, A091562, A205575 (same recurrence).
a(n, k) = A090174(n-1, k), k

T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k) + T(n-2, k-1) + T(n-2, k-2) for n >= 2, k >= 0, with initial conditions specified by first two rows.

Edited and extended by Christian G. Bower, Jan 20 2004

A091562 Triangle read by rows, related to Pascal's triangle, starting with 1, 0, 0.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 1, 2, 2, 1, 2, 5, 7, 5, 2, 3, 10, 17, 17, 10, 3, 5, 20, 41, 51, 41, 20, 5, 8, 38, 91, 136, 136, 91, 38, 8, 13, 71, 195, 339, 405, 339, 195, 71, 13, 21, 130, 403, 799, 1107, 1107, 799, 403, 130, 21, 34, 235, 812, 1807, 2845, 3297, 2845, 1807, 812, 235, 34

Offset: 0

Christian G. Bower, Jan 20 2004

			Triangle begins:
  1;
  0,0;
  1,1,1;
  1,2,2,1;
  2,5,7,5,2;
  ...

Row sums: A054878, column 0: A000045(n-1), column 1: A001629.
Cf. A090171, A090172, A090173, A090174, A091533, A205575 (same recurrence).
Cf. A090172.

T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k) + T(n-2, k-1) + T(n-2, k-2) for n >= 2, k >= 0, with initial conditions specified by first two rows.

G.f.: A(x, y) = (1-x-x*y)/(1-x-x*y-x^2-x^2*y-x^2*y^2).

A090173 Triangle read by rows, related to Pascal's triangle read mod 2, starting with 0, 0, 1.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jan 19 2004

			Triangle begins
  0;
  0,1;
  0,1,1;
  0,0,1,0;
  0,1,1,1,1;
  ...

Y. Moshe, The density of 0's in recurrence double sequences, J. Number Theory, 103 (2003), 109-121; see Fig. 2.

Cf. A007318, A083093.
Cf. A090171, A090172, A090174, A091533, A091562, A205575 (same recurrence).
a(n, k) = A090174(n-1, k-1), k>0, 0 otherwise.

T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k) + T(n-2, k-1) + T(n-2, k-2) for n >= 2, k >= 0, with initial conditions specified by first two rows.

Edited and extended by Christian G. Bower, Jan 20 2004

A205575 Triangle read by rows, related to Pascal's triangle, starting with rows 1; 1,0.

Original entry on oeis.org

1, 1, 0, 2, 2, 1, 3, 5, 4, 1, 5, 12, 14, 8, 2, 8, 25, 38, 32, 15, 3, 13, 50, 94, 104, 71, 28, 5, 21, 96, 215, 293, 260, 149, 51, 8, 34, 180, 468, 756, 822, 612, 304, 92, 13, 55, 331, 980, 1828, 2346, 2136, 1376, 604, 164, 21

Offset: 0

Philippe Deléham, Jan 29 2012

Antidiagonal sums are in A052980, row sums are in A046717.

Similar to A091533 and to A091562. Triangle satisfying the same recurrence as A091533 and A091562, but with the initial values T(0,0) = 1, T(0,1) = 1, T(1,1) = 0.

			Triangle begins :
1
1, 0
2, 2, 1
3, 5, 4, 1
5, 12, 14, 8, 2
8, 25, 38, 32, 15, 3
13, 50, 94, 104, 71, 28, 5

Cf. Column 0: A000045, Diagonals : A000045, A029907, A036681.

Cf. A090171, A090172, A090173, A090174, A091533, A091562 (same recurrence).

T(n,k) = {if(n<0, return(0)); if (n==0, if (k<0, return(0)); if (k==0, return(1))); if (n==1, if (k<0, return(0)); if (k==0, return(1)); if (k==1, return(0))); T(n-1,k)+T(n-1,k-1)+T(n-2,k)+T(n-2,k-1)+T(n-2,k-2);} \\ Michel Marcus, Oct 27 2021

T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + T(n-2,k-2) for n>=2, k>=0, with initial conditions specified by first two rows. T(0,0) = 1, T(1,0) = 1, T(1,1) = 0.

a(46), a(48) corrected by Georg Fischer, Oct 27 2021

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A091563 Row sums of A090172.

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A090171 Triangle read by rows, related to Pascal's triangle read mod 2, starting with 0, 1, 0.

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A091562 Triangle read by rows, related to Pascal's triangle, starting with 1, 0, 0.

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A090173 Triangle read by rows, related to Pascal's triangle read mod 2, starting with 0, 0, 1.

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A205575 Triangle read by rows, related to Pascal's triangle, starting with rows 1; 1,0.

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