A089597 -id:A089597 - cp's OEIS frontend

A014670 G.f.: (1+x)(1+x^3)(1+x^5)(1+x^7)(1+x^9)/((1-x^2)(1-x^4)(1-x^6)(1-x^8)(1-x^10)).

1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 16, 20, 26, 32, 38, 47, 58, 69, 81, 96, 114, 133, 153, 177, 206, 236, 267, 304, 346, 390, 437, 490, 550, 613, 679, 753, 835, 921, 1011, 1111, 1221, 1335, 1455, 1586, 1728, 1877, 2032, 2200, 2382, 2571, 2768, 2980, 3207, 3443, 3689, 3952

Offset: 0

N. J. A. Sloane, Dec 31 2003

nonn

Poincaré series [or Poincare series] (or Molien series) for symmetric invariants in F_2(b_1, b_2, ... b_n) ⊗ E(e_1, e_2, ... e_n) with b_i 2-dimensional, e_i one-dimensional and the permutation action of S_n, in the case n=5.

A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 108.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3, -5, 8, -11, 14, -18, 21, -23, 24, -24, 23, -21, 18, -14, 11, -8, 5, -3, 1).

Cf. A006950, A000933, A089597.

CoefficientList[Series[(1+x)*(1+x^3)*(1+x^5)*(1+x^7)*(1+x^9)/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^8)*(1-x^10)), {x, 0, 50}], x] (* Jinyuan Wang, Mar 10 2020 *)
LinearRecurrence[{3,-5,8,-11,14,-18,21,-23,24,-24,23,-21,18,-14,11,-8,5,-3,1},{1,1,1,2,3,4,5,7,10,13,16,20,26,32,38,47,58,69,81},60] (* Harvey P. Dale, Mar 28 2023 *)

Vec((1+x)*(1+x^3)*(1+x^5)*(1+x^7)*(1+x^9)/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^8)*(1-x^10))+ O(x^100)) \\ Michel Marcus, Mar 18 2014

G.f.: -(x^2-x+1) *(x^6-x^5+x^4-x^3+x^2-x+1) *(x^6-x^3+1) / ( (x^4+x^3+x^2+x+1) *(1+x+x^2) *(x^4+1) *(x^2+1)^2 *(x-1)^5 ). - R. J. Mathar, Dec 18 2014

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.

Original entry on oeis.org

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)).

A316718 Expansion of Product_{k=1..6} (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, 34, 41, 52, 65, 78, 93, 113, 137, 162, 189, 224, 266, 308, 355, 414, 480, 549, 626, 717, 820, 928, 1045, 1183, 1337, 1496, 1670, 1871, 2091, 2321, 2571, 2853, 3161, 3484, 3830, 4218, 4640, 5078, 5549, 6072, 6633, 7219

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), this sequence (b=6), A316719 (b=7), A316720 (b=8), A316721 (b=9), A316722 (b=10).

Cf. A316675.

N=99; x='x+O('x^N); Vec(prod(k=1, 6, (1+x^(2*k-1))/(1-x^(2*k))))

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

A014670 G.f.: (1+x)*(1+x^3)*(1+x^5)*(1+x^7)*(1+x^9)/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^8)*(1-x^10)).

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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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A316718 Expansion of Product_{k=1..6} (1+x^(2*k-1))/(1-x^(2*k)).

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

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

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

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

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A014670 G.f.: (1+x)(1+x^3)(1+x^5)(1+x^7)(1+x^9)/((1-x^2)(1-x^4)(1-x^6)(1-x^8)(1-x^10)).

A316718 Expansion of Product_{k=1..6} (1+x^(2k-1))/(1-x^(2k)).

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)).