A316384 - OEIS

%I #123 Jul 29 2023 05:04:21

%S 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,

%T 7,17,7,19,8,22,10,26,12,30,13,33,14,38,17,45,21,51,22,56,24,64,29,74,

%U 33,83,36,92,40,104,46,119,53,133,58,147,63,165,73,187,83,208,90

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

%C            *

%C           / \

%C        *-*-*-*-*

%C         \ / \ /

%C          *---*

%C           \ /

%C            *

%C Such a way to stack is not allowed.

%C From _George Beck_, Jul 28 2023: (Start)

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

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

%C    2 2 2 2 1

%C    2

%C    2

%C    2

%C    1

%C (End)

%H Seiichi Manyama, <a href="/A316384/b316384.txt">Table of n, a(n) for n = 0..10000</a>

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

%F a(2n  ) = A036016(n).

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

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

%F a(n) ~ sqrt(sqrt(2) + (-1)^n) * exp(Pi*sqrt(n)/2^(3/2)) / (4*n^(3/4)). - _Vaclav Kotesovec_, Feb 08 2023

%F 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

%e a(16) = 4.

%e                                  *   *

%e                                 / \ / \

%e      *---*---*---*---*         *---*---*

%e       \ / \ / \ / \ /         / \ / \ / \

%e        *---*---*---*         *---*---*---*

%e         \ / \ / \ /           \ / \ / \ /

%e          *---*---*             *---*---*

%e           \ / \ /               \ / \ /

%e            *---*                 *---*

%e             \ /                   \ /

%e              *                     *

%e    *---*           *---*     *           *

%e     \ / \         / \ /     / \         / \

%e      *---*       *---*     *---*   *   *---*

%e       \ / \     / \ /       \ / \ / \ / \ /

%e        *---*   *---*         *---*---*---*

%e         \ / \ / \ /           \ / \ / \ /

%e          *---*---*             *---*---*

%e           \ / \ /               \ / \ /

%e            *---*                 *---*

%e             \ /                   \ /

%e              *                     *

%e a(17) = 2.

%e            *---*         *---*           *---*

%e           / \ / \         \ / \         / \ /

%e          *---*---*         *---*       *---*

%e         / \ / \ / \         \ / \     / \ /

%e        *---*---*---*         *---*---*---*

%e         \ / \ / \ /           \ / \ / \ /

%e          *---*---*             *---*---*

%e           \ / \ /               \ / \ /

%e            *---*                 *---*

%e             \ /                   \ /

%e              *                     *

%t 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 *)

%o (Ruby)

%o def s(k, n)

%o   s = 0

%o   (1..n).each{|i| s += i if n % i == 0 && i % k == 0}

%o   s

%o end

%o def A(ary, n)

%o   a_ary = [1]

%o   a = [0] + (1..n).map{|i| ary.inject(0){|s, j| s + j[1] * s(j[0], i)}}

%o   (1..n).each{|i| a_ary << (1..i).inject(0){|s, j| s - a[j] * a_ary[-j]} / i}

%o   a_ary

%o end

%o def A316384(n)

%o   A([[1, 1], [4, -1]], n).map{|i| i.abs}

%o end

%o p A316384(100)

%Y 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.

%K nonn

%O 0,9

%A _Seiichi Manyama_, Jun 30 2018