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

0,9

Comments

*

/ \

*-*-*-*-*

\ / \ /

*---*

\ /

*

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)

Links

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

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

Example

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

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 *)

Prog

(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)

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.

Sequence in context: A025825 A293224 A082303 * A029838 A213649 A242397

Adjacent sequences: A316381 A316382 A316383 * A316385 A316386 A316387

Keyword

nonn

Author

Seiichi Manyama, Jun 30 2018

Status

approved