A316718 -id:A316718 - cp's OEIS frontend

A316675 Triangle read by rows: T(n,k) gives the number of ways to stack n triangles in a valley so that the right wall has k triangles for n >= 0 and 0 <= k <= n.

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 2, 1, 1, 1, 0, 0, 1, 1, 3, 2, 1, 1, 1, 0, 0, 1, 1, 3, 3, 2, 1, 1, 1, 0, 0, 1, 1, 3, 3, 3, 2, 1, 1, 1, 0, 0, 1, 1, 4, 3, 4, 3, 2, 1, 1, 1, 0, 0, 1, 1, 5, 4, 5, 4, 3, 2, 1, 1, 1

Offset: 0

Seiichi Manyama, Jul 10 2018

			T(8,4) = 3.
    *                             *
   / \                           / \
  *---*   *     *---*---*       *---*
   \ / \ / \     \ / \ / \     / \ / \
    *---*---*     *---*---*   *---*---*
     \ / \ /       \ / \ /     \ / \ /
      *---*         *---*       *---*
       \ /           \ /         \ /
        *             *           *
Triangle begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 1, 1;
  0, 0, 1, 1, 1;
  0, 0, 1, 1, 1, 1;
  0, 0, 1, 1, 1, 1,  1;
  0, 0, 1, 1, 2, 1,  1,  1;
  0, 0, 1, 1, 3, 2,  1,  1,  1;
  0, 0, 1, 1, 3, 3,  2,  1,  1,  1;
  0, 0, 1, 1, 3, 3,  3,  2,  1,  1,  1;
  0, 0, 1, 1, 4, 3,  4,  3,  2,  1,  1, 1;
  0, 0, 1, 1, 5, 4,  5,  4,  3,  2,  1, 1, 1;
  0, 0, 1, 1, 5, 5,  6,  5,  4,  3,  2, 1, 1, 1;
  0, 0, 1, 1, 5, 5,  8,  6,  5,  4,  3, 2, 1, 1, 1;
  0, 0, 1, 1, 6, 5, 10,  8,  7,  5,  4, 3, 2, 1, 1, 1;
  0, 0, 1, 1, 7, 6, 11, 10, 10,  7,  5, 4, 3, 2, 1, 1, 1;
  0, 0, 1, 1, 7, 7, 13, 11, 12, 10,  7, 5, 4, 3, 2, 1, 1, 1;
  0, 0, 1, 1, 7, 7, 16, 13, 14, 12, 10, 7, 5, 4, 3, 2, 1, 1, 1;
  ...

Seiichi Manyama, Rows n = 0..100, flattened

Row sums give A006950.
Sums of even columns give A059777.
Cf. A004525, A000933, A089597, A014670, A316718, A316719, A316720, A316721, A316722.
Cf. A072233.

For m >= 0,
Sum_{n>=2m}   T(n,2m)  *x^n = x^(2m)   * Product_{j=1..m} (1+x^(2j-1))/(1-x^(2j)).
Sum_{n>=2m+1} T(n,2m+1)*x^n = x^(2m+1) * Product_{j=1..m} (1+x^(2j-1))/(1-x^(2j)).

A316719 Expansion of Product_{k=1..7} (1+x^(2k-1))/(1-x^(2k)).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 16, 21, 28, 35, 43, 54, 68, 83, 100, 122, 149, 179, 212, 253, 303, 357, 417, 490, 575, 668, 772, 893, 1033, 1187, 1356, 1551, 1773, 2015, 2281, 2583, 2922, 3291, 3695, 4147, 4650, 5197, 5791, 6450, 7179, 7966, 8818, 9757, 10785, 11893

Offset: 0

Seiichi Manyama, Jul 11 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Product_{k=1..b} (1+x^(2*k-1))/(1-x^(2*k)): A000012 (b=1), A004525(n+1) (b=2), A000933(n+5) (b=3), A089597 (b=4), A014670 (b=5), A316718 (b=6), this sequence (b=7), A316720 (b=8), A316721 (b=9), A316722 (b=10).

Cf. A316675.

nmax=60; CoefficientList[Series[Product[(1 + x^(2 k - 1)) / (1 - x^(2 k)), {k, 1, 7}], {x, 0, nmax}], x] (* Vincenzo Librandi, Jul 12 2018 *)

N=99; x='x+O('x^N); Vec(prod(k=1, 7, (1+x^(2*k-1))/(1-x^(2*k))))

A316720 Expansion of Product_{k=1..8} (1+x^(2k-1))/(1-x^(2k)).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 16, 21, 28, 35, 43, 55, 70, 85, 103, 127, 156, 188, 224, 270, 326, 386, 454, 539, 638, 746, 869, 1016, 1186, 1372, 1581, 1827, 2108, 2415, 2758, 3156, 3605, 4094, 4639, 5261, 5956, 6715, 7553, 8499, 9552, 10694, 11950, 13357, 14908, 16589

Offset: 0

Seiichi Manyama, Jul 11 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Product_{k=1..b} (1+x^(2*k-1))/(1-x^(2*k)): A000012 (b=1), A004525(n+1) (b=2), A000933(n+5) (b=3), A089597 (b=4), A014670 (b=5), A316718 (b=6), A316719 (b=7), this sequence (b=8), A316721 (b=9), A316722 (b=10).

Cf. A316675.

N=99; x='x+O('x^N); Vec(prod(k=1, 8, (1+x^(2*k-1))/(1-x^(2*k))))

A316721 Expansion of Product_{k=1..9} (1+x^(2k-1))/(1-x^(2k)).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 16, 21, 28, 35, 43, 55, 70, 86, 105, 129, 159, 193, 231, 279, 338, 403, 477, 568, 675, 795, 932, 1094, 1284, 1497, 1736, 2016, 2340, 2700, 3105, 3573, 4106, 4699, 5363, 6118, 6972, 7921, 8974, 10163, 11500, 12974, 14606, 16435, 18471

Offset: 0

Seiichi Manyama, Jul 11 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Product_{k=1..b} (1+x^(2*k-1))/(1-x^(2*k)): A000012 (b=1), A004525(n+1) (b=2), A000933(n+5) (b=3), A089597 (b=4), A014670 (b=5), A316718 (b=6), A316719 (b=7), A316720 (b=8), this sequence (b=9), A316722 (b=10).

Cf. A316675.

nmax=50; CoefficientList[Series[Product[(1 + x^(2 k - 1)) / (1 - x^(2 k)), {k, 1, 9}], {x, 0, nmax}], x] (* Vincenzo Librandi, Jul 12 2018 *)

N=99; x='x+O('x^N); Vec(prod(k=1, 9, (1+x^(2*k-1))/(1-x^(2*k))))

A316722 Expansion of Product_{k=1..10} (1+x^(2k-1))/(1-x^(2k)).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 16, 21, 28, 35, 43, 55, 70, 86, 105, 130, 161, 195, 234, 284, 345, 412, 489, 585, 698, 824, 969, 1143, 1347, 1575, 1834, 2141, 2496, 2891, 3339, 3862, 4460, 5125, 5876, 6740, 7720, 8810, 10031, 11423, 12993, 14730, 16669, 18862, 21315

Offset: 0

Seiichi Manyama, Jul 11 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Product_{k=1..b} (1+x^(2*k-1))/(1-x^(2*k)): A000012 (b=1), A004525(n+1) (b=2), A000933(n+5) (b=3), A089597 (b=4), A014670 (b=5), A316718 (b=6), A316719 (b=7), A316720 (b=8), A316721 (b=9), this sequence (b=10).

Cf. A316675.

N=99; x='x+O('x^N); Vec(prod(k=1, 10, (1+x^(2*k-1))/(1-x^(2*k))))

cp's OEIS Frontend

A316675 Triangle read by rows: T(n,k) gives the number of ways to stack n triangles in a valley so that the right wall has k triangles for n >= 0 and 0 <= k <= n.

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A316719 Expansion of Product_{k=1..7} (1+x^(2*k-1))/(1-x^(2*k)).

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PARI

A316720 Expansion of Product_{k=1..8} (1+x^(2*k-1))/(1-x^(2*k)).

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PARI

A316721 Expansion of Product_{k=1..9} (1+x^(2*k-1))/(1-x^(2*k)).

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Mathematica

PARI

A316722 Expansion of Product_{k=1..10} (1+x^(2*k-1))/(1-x^(2*k)).

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PARI

A316719 Expansion of Product_{k=1..7} (1+x^(2k-1))/(1-x^(2k)).

A316720 Expansion of Product_{k=1..8} (1+x^(2k-1))/(1-x^(2k)).

A316721 Expansion of Product_{k=1..9} (1+x^(2k-1))/(1-x^(2k)).

A316722 Expansion of Product_{k=1..10} (1+x^(2k-1))/(1-x^(2k)).