A316384 Number of ways to stack n triangles symmetrically in a valley (pointing upwards or downwards depending on row parity).

1, 1, 1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 4, 2, 5, 2, 5, 2, 6, 3, 8, 4, 9, 4, 10, 4, 12, 6, 15, 7, 17, 7, 19, 8, 22, 10, 26, 12, 30, 13, 33, 14, 38, 17, 45, 21, 51, 22, 56, 24, 64, 29, 74, 33, 83, 36, 92, 40, 104, 46, 119, 53, 133, 58, 147, 63, 165, 73, 187, 83, 208, 90

Offset: 0

Author

Seiichi Manyama, Jun 30 2018

Keywords

• nonn

Comments

*

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*-*-*-*-*

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Such a way to stack is not allowed.

From George Beck, Jul 28 2023: (Start)

Equivalently, a(n) is the number of partitions of n such that the 2-modular Ferrers diagram is symmetric.

The first example for n = 16 below corresponds to the partition 9 + 2 + 2 + 2 + 1 with 2-modular Ferrers diagram:

2 2 2 2 1

2

2

2

1

(End)

Examples

			a(16) = 4.
                                 *   *
                                / \ / \
     *---*---*---*---*         *---*---*
      \ / \ / \ / \ /         / \ / \ / \
       *---*---*---*         *---*---*---*
        \ / \ / \ /           \ / \ / \ /
         *---*---*             *---*---*
          \ / \ /               \ / \ /
           *---*                 *---*
            \ /                   \ /
             *                     *
   *---*           *---*     *           *
    \ / \         / \ /     / \         / \
     *---*       *---*     *---*   *   *---*
      \ / \     / \ /       \ / \ / \ / \ /
       *---*   *---*         *---*---*---*
        \ / \ / \ /           \ / \ / \ /
         *---*---*             *---*---*
          \ / \ /               \ / \ /
           *---*                 *---*
            \ /                   \ /
             *                     *
a(17) = 2.
           *---*         *---*           *---*
          / \ / \         \ / \         / \ /
         *---*---*         *---*       *---*
        / \ / \ / \         \ / \     / \ /
       *---*---*---*         *---*---*---*
        \ / \ / \ /           \ / \ / \ /
         *---*---*             *---*---*
          \ / \ /               \ / \ /
           *---*                 *---*
            \ /                   \ /
             *                     *
		

Links

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

Crossrefs

Cf. A000700 (number of symmetric Ferrers graphs with n nodes), A006950 (number of ways to stack n triangles in a valley), A029838, A036015, A036016, A082303.

Programs

Mathematica

nmax = 100; CoefficientList[Series[(QPochhammer[x^6, x^16]*QPochhammer[x^10, x^16] + x*QPochhammer[x^2, x^16]*QPochhammer[x^14, x^16])/(QPochhammer[x^2, x^4] * QPochhammer[x^8, x^16]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 08 2023 *)

Ruby

def s(k, n)
  s = 0
  (1..n).each{|i| s += i if n % i == 0 && i % k == 0}
  s
end
def A(ary, n)
  a_ary = [1]
  a = [0] + (1..n).map{|i| ary.inject(0){|s, j| s + j[1] * s(j[0], i)}}
  (1..n).each{|i| a_ary << (1..i).inject(0){|s, j| s - a[j] * a_ary[-j]} / i}
  a_ary
end
def A316384(n)
  A([[1, 1], [4, -1]], n).map{|i| i.abs}
end
p A316384(100)

Formula

a(2n+1) = A036015(n).

a(2n ) = A036016(n).

a(n) = |A029838(n)| = |A082303(n)|.

Euler transform of period 16 sequence [1, 0, -1, 1, -1, 1, 1, -1, 1, 1, -1, 1, -1, 0, 1, 0, ...].

a(n) ~ sqrt(sqrt(2) + (-1)^n) * exp(Pi*sqrt(n)/2^(3/2)) / (4*n^(3/4)). - Vaclav Kotesovec, Feb 08 2023

G.f.: Product_{k>=1} 1/((1 - x^(16*k-2))*(1 - x^(16*k-8))*(1 - x^(16*k-14))) + x*Product_{k>=1} 1/((1 - x^(16*k-6))*(1 - x^(16*k-8))*(1 - x^(16*k-10))). - Vaclav Kotesovec, Feb 08 2023