A129686 -id:A129686 - cp's OEIS frontend

A131084 A129686 * A007318. Riordan triangle (1+x, x/(1-x)).

1, 1, 1, 0, 2, 1, 0, 2, 3, 1, 0, 2, 5, 4, 1, 0, 2, 7, 9, 5, 1, 0, 2, 9, 16, 14, 6, 1, 0, 2, 11, 25, 30, 20, 7, 1, 0, 2, 13, 36, 55, 50, 27, 8, 1, 0, 2, 15, 49, 91, 105, 77, 35, 9, 1

Offset: 1

Gary W. Adamson, Jun 14 2007

Row sums = A098011 starting (1, 2, 3, 6, 12, 24, 48, ...). A131085 = A007318 * A129686

Riordan array (1+x, x/(1-x)). - Philippe Deléham, Mar 02 2012

			The triangle T(n, k) begins:
n\k 0  1  2  3   4   5   6   7  8  9 10 ...
0:  1
1:  1  1
2:  0  2  1
3:  0  2  3  1
4:  0  2  5  4   1
5:  0  2  7  9   5   1
6:  0  2  9 16  14   6   1
7:  0  2 11 25  30  20   7   1
8:  0  2 13 36  55  50  27   8  1
9:  0  2 15 49  91 105  77  35  9  1
10: 0  2 17 64 140 196 182 112 44 10  1
... Reformatted. - _Wolfdieter Lang_, Jan 06 2015

Cf. A007318, A129686, A098011, A131085.

Cf. A029653, A131084, A208510.

A129686(signed): (1,1,1,...) in the main diagonal and (-1,-1,-1, ...) in the subsubdiagonal); * A007318, Pascal's triangle; as infinite lower triangular matrices.
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(2*x + 3*x^2/2! + x^3/3!) = 2*x + 7*x^2/2! + 16*x^3/3! + 30*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
G.f. column k: (1+x)*(x/(1-x))^k, k >= 0. (Riordan property). - Wolfdieter Lang, Jan 06 2015
T(n, 0) = 1 if n=0 or n=1 else 0; T(n, k) = binomial(n-1,k-1) + binomial(n-2,k-1)*[n-1 >= k] if n >= k >= 1, where [S] = 1 if S is true, else 0, and  T(n, k) = 0 if n < k. - Wolfdieter Lang, Jan 08 2015

Edited: Added Riordan property (see Philippe Deléham comment) in name. - Wolfdieter Lang, Jan 06 2015

A129687 A129686 * A007318.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 4, 3, 1, 2, 6, 7, 4, 1, 2, 8, 13, 11, 5, 1, 2, 10, 21, 24, 16, 6, 1, 2, 12, 31, 45, 40, 22, 7, 1, 2, 14, 43, 76, 85, 62, 29, 8, 1, 2, 16, 57, 119, 161, 147, 91, 37, 9, 1, 2, 18, 73, 176, 280, 308, 238, 128, 46, 10, 1, 2, 20, 91, 249, 456

Offset: 0

Gary W. Adamson, Apr 28 2007

Row sums = A084215: (1, 2, 5, 10, 20, 40, 80, ...). A007318 * A129686 = A124725.
From Philippe Deléham, Feb 12 2014: (Start)
Riordan array ((1+x^2)/(1-x), x/(1-x)).
Diagonal sums are A000032(n) - 0^n (cf. A000204).
T(n,0) = A046698(n+1).
T(n+1,1) = A004277(n).
T(n+2,2) = A002061(n+1).
T(n+3,3) = A006527(n+1) = A167875(n).
T(n+4,4) = A006007(n+1).
T(n+5,5) = A081282(n+1). (End)

			First few rows of the triangle:
  1;
  1,   1;
  2,   2,   1;
  2,   4,   3,   1;
  2,   6,   7,   4,   1;
  2,   8,  13,  11,   5,   1;
  2,  10,  21,  24,  16,   6,   1;
  2,  12,  31,  45,  40,  22,   7,   1;
  2,  14,  43,  76,  85,  62,  29,   8,   1;
  2,  16,  57, 119, 161, 147,  91,  37,   9,   1;
  ...

Cf. A129686, A007318, A124725.

Cf. Columns: A046698, A004277, A002061, A006527, A006007, A081282

A129686 * A007318 (Pascal's Triangle), as infinite lower triangular matrices.

T(n,k) = T(n-1,k) + T(n-1,k-1), T(0,0) = T(1,0) = T(1,1) = T(2,2) = 1, T(2,0) = T(2,1) = 2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Feb 12 2014

More terms from Philippe Deléham, Feb 12 2014

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.

A131034 A129686 * A051340.

Original entry on oeis.org

1, 2, 1, 4, 1, 1, 6, 2, 1, 1, 8, 2, 2, 1, 1, 10, 2, 2, 2, 1, 12, 2, 2, 2, 2, 1, 1, 14, 2, 2, 2, 2, 2, 1, 1, 16, 2, 2, 2, 2, 2, 2, 1, 1

Offset: 0

Gary W. Adamson, Jun 10 2007

Row sums = A113127: (1, 3, 6, 10, 14, 18, 22, ...).

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

Cf. A129686, A051340, A113127.

A129686 * A051340 as infinite lower triangular matrices.

A131085 Triangle T(n,k) (n>=0, 0<=k<=n-1) read by rows, A007318 * A129686.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, -2, 2, 3, 1, -5, 0, 5, 4, 1, -9, -5, 5, 9, 5, 1, -14, -14, 0, 14, 14, 6, 1, -20, -28, -14, 14, 28, 20, 7, 1, -27, -48, -42, 0, 42, 48, 27, 8, 1, -35, -75, -90, -42, 42, 90, 75, 35, 9, 1, -44, -110, -165, -132, 0, 132, 165, 110, 44, 10, 1

Offset: 0

Gary W. Adamson, Jun 14 2007

Row sums = n.

A131085 * A000012 = A074909 starting (1, 2, 1, 3, 3, ...) instead of (1, 1, 2, 1, 3, 3, ...).

			First few rows of the triangle are:
   1;
   1,  1;
   0,  2, 1;
  -2,  2, 3, 1;
  -5,  0, 5, 4, 1;
  -9, -5, 5, 9, 5, 1;
-14, -14, 0, 14, 14, 6, 1;
-20, -28, -14, 14, 28, 20, 7, 1;
-27, -48, -42, 0, 42, 48, 27, 8, 1;
-35, -75, -90, -42, 42, 90, 75, 35, 9, 1;
   ...

Cf. A007318, A009766, A097808, A112466, A214292, A112467, A129686, A131084.

tabl(nn) = {t007318 = matrix(nn, nn, n, k, binomial(n-1, k-1)); t129686 = matrix(nn, nn, n, k, (k<=n)*((-1)^((n-k)\2)*((k==n) || (-1)*(k==(n-2))))); t131085 = t007318*t129686; for (n = 1, nn, for (k = 1, n, print1(t131085[n, k], ", ");););} \\ Michel Marcus, Feb 12 2014

Binomial transform of A129686 signed with (1, 1, 1, ...) in the main diagonal and (-1, -1, -1, ...) in the subsubdiagonal.
T(n,m) = T(n-1,m-1) + T(n-1,m). - Yuchun Ji, Dec 17 2018
T(2*k,k-1) = 0 for k > 0. - Yuchun Ji, Dec 20 2018
Comparing this triangle with the Catalan triangle A009766 one can see many similarities. For example, T(k+j,k) = A009766(k+1,j) for j < k+2. - Yuchun Ji, Dec 23 2018 [Edited by N. J. A. Sloane, Feb 11 2019]

Missing comma corrected by Naruto Canada, Aug 26 2007
More terms from Michel Marcus, Feb 12 2014
Offset changed by N. J. A. Sloane, Feb 11 2019

A133938 A007318 * (A129686 + A133080 - I), where I is the identity matrix.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 7, 4, 4, 1, 11, 8, 11, 4, 1, 16, 15, 25, 11, 6, 1, 22, 26, 50, 26, 22, 6, 1, 29, 42, 91, 56, 63, 22, 8, 1, 37, 64, 154, 112, 154, 64, 37, 8, 1, 46, 93, 24, 210, 336, 162, 129, 37, 10, 1

Offset: 1

Gary W. Adamson, Sep 29 2007

Left column = A000124: (1, 2, 4, 7, 11, ...).

Row sums = A133124: (1, 3, 7, 16, 35, 74, ...).

			First few rows of the triangle:
   1;
   2,  1;
   4,  2,  1;
   7,  4,  4,  1;
  11,  8, 11,  4,  1;
  16, 15, 25, 11,  6,  1;
  22, 26, 50, 26, 22,  6,  1;
  ...

Cf. A000124, A133124, A129686, A133080.

Binomial transform of matrix M, where M = a tridiagonal matrix with (1,1,1,...) in the main diagonal, (1,0,1,0,...) in the subdiagonal and (1,1,1,...) in the subsubdiagonal. M = (A129686 + A133080 - I), I = Identity matrix.

A130028 A129686 * A054523.

Original entry on oeis.org

1, 1, 1, 3, 0, 1, 3, 2, 0, 1, 6, 0, 1, 0, 1, 4, 3, 1, 1, 0, 1, 10, 0, 0, 0, 1, 0, 1, 6, 4, 1, 1, 0, 1, 0, 1, 12, 0, 2, 0, 0, 0, 1, 0, 1, 8, 6, 0, 1, 1, 0, 0, 1, 0, 1

Offset: 1

Gary W. Adamson, May 02 2007

Row sums = A004277, 1 followed by even terms: (1, 2, 4, 6, 8, 10, ...).

			First few rows of the triangle:
   1;
   1, 1;
   3, 0, 1;
   3, 2, 0, 1;
   6, 0, 1, 0, 1;
   4, 3, 1, 1, 0, 1;
  10, 0, 0, 0, 1, 0, 1;
   6, 4, 1, 1, 0, 1, 0, 1;
  ...

Cf. A129686, A054523, A004277.

A129686 * A054523 as infinite lower triangular matrices, where A129686 = the alternate term operator.

A131035 A051340 * A129686 as infinite lower triangular matrices.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Jun 10 2007

Row sums give A008486.

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

Cf. A051340, A129686, A008486, A131034.

A131258 A129686^(-1) * A052509.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 0, 2, 2, 1, 1, 2, 3, 2, 1, 1, 3, 5, 3, 2, 1, 0, 4, 8, 6, 3, 2, 1, 0, 4, 11, 12, 6, 3, 2, 1, 1, 4, 14, 20, 13, 6, 3, 2, 1, 1, 5, 18, 30, 25, 13, 6, 3, 2, 1

Offset: 0

Gary W. Adamson, Jun 23 2007

Row sums = A097083: (1, 2, 3, 5, 9, 15, ...).

			First few rows of the triangle:
  1;
  1, 1;
  0, 2, 1;
  0, 2, 2, 1;
  1, 2, 3, 2, 1;
  1, 3, 5, 3, 2, 1;
  0, 4, 8, 6, 3, 2, 1;
  ...

Cf. A129686, A052509, A097083.

A129686^(-1) * A052509 as infinite lower triangular matrices, where A129686 = the alternate term operator.

cp's OEIS Frontend

A131084 A129686 * A007318. Riordan triangle (1+x, x/(1-x)).

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A129687 A129686 * A007318.

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A129688 A129686 * A128174.

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A131034 A129686 * A051340.

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A131085 Triangle T(n,k) (n>=0, 0<=k<=n-1) read by rows, A007318 * A129686.

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A133938 A007318 * (A129686 + A133080 - I), where I is the identity matrix.

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A130028 A129686 * A054523.

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A131035 A051340 * A129686 as infinite lower triangular matrices.

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A131258 A129686^(-1) * A052509.

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