A036015 -id:A036015 - cp's OEIS frontend

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

Seiichi Manyama, Jun 30 2018

nonn

           *
          / \
       *-*-*-*-*
        \ / \ /
         *---*
          \ /
           *
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)

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

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

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.

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 *)

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)

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

A036016 Number of partitions of n into parts not of form 4k+2, 8k, 8k+3 or 8k-3.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 4, 5, 5, 6, 8, 9, 10, 12, 15, 17, 19, 22, 26, 30, 33, 38, 45, 51, 56, 64, 74, 83, 92, 104, 119, 133, 147, 165, 187, 208, 229, 256, 288, 319, 351, 390, 435, 481, 528, 584, 649, 715, 783, 863, 954, 1047, 1145, 1258, 1385, 1517, 1655, 1812, 1989

Offset: 0

Olivier Gérard

Case k=2,i=2 of Gordon/Goellnitz/Andrews Theorem.
Also number of partitions in which no odd part is repeated, with at most one part of size less than or equal to 2 and where differences between adjacent parts are greater than 1 when the larger part is odd and greater than 2 when the larger part is even.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 4*x^8 + 5*x^9 + 5*x^10 + ...

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 114.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
G. E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573. See Th. 8.
S.-D. Chen and S.-S. Huang, On the series expansion of the Göllnitz-Gordon continued fraction, Internat. J. Number Theory, 1 (2005), 53-63.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
Nicolas Allen Smoot, A Partition Function Connected with the Göllnitz--Gordon Identities, arXiv:2005.09263 [math.NT], 2020. See g1(n) Table 1 p. 22.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Goellnitz-Gordon Identities

Cf. A036015, A316384.

M:=100; qf:=(a,q)->mul(1-a*q^j,j=0..M); tS:=1/(qf(q,q^8)*qf(q^4,q^8)*qf(q^7,q^8)); series(%,q,M); seriestolist(%);

nmax=60; CoefficientList[Series[Product[1/((1-x^(8*k-1))*(1-x^(8*k-4))*(1-x^(8*k-7))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 04 2015 *)

{a(n) = if( n<0, 0, polcoeff( 1 / prod( k=1, n, 1 - ([1, 0, 0, 1, 0, 0, 1, 0][(k-1)%8 + 1]) * x^k, 1 + x * O(x^n)), n))} /* Michael Somos, Jun 28 2004 */

Expansion of f(-x^3, -x^5) / psi(-x) = psi(x^4) / f(-x, -x^7) in powers of x where phi(), f(,) are Ramanujan theta functions.
Euler transform of period 8 sequence [ 1, 0, 0, 1, 0, 0, 1, 0, ...]. - Michael Somos, Jun 28 2004
Let qf(a, q) = Product(1-a*q^j, j=0..infinity); g.f. is 1/(qf(q, q^8)*qf(q^4, q^8)*qf(q^7, q^8)).
G.f.: Sum_{k>=0} x^(k^2) Product_{i=1..k} (1 + x^(2*i - 1)) / (1 - x^(2*i)). - Michael Somos, Jul 24 2012
a(n) ~ sqrt(2+sqrt(2)) * exp(sqrt(n)*Pi/2) / (8*n^(3/4)). - Vaclav Kotesovec, Oct 04 2015

A259774 Expansion of f(x, x^7) / f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 0, 0, -1, 1, -1, 1, -1, 2, -2, 2, -3, 4, -4, 4, -6, 7, -7, 8, -10, 12, -13, 14, -17, 21, -22, 24, -29, 33, -36, 40, -46, 53, -58, 63, -73, 83, -90, 99, -113, 127, -138, 152, -171, 191, -209, 228, -255, 285, -309, 338, -377, 416, -453, 495, -547, 603, -656

Offset: 0

Michael Somos, Nov 08 2015

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 - x^3 + x^4 - x^5 + x^6 - x^7 + 2*x^8 - 2*x^9 + 2*x^10 - 3*x^11 + ...
G.f. = q^7 - q^55 + q^71 - q^87 + q^103 - q^119 + 2*q^135 - 2*q^151 + ...

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 15th equation.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A029838, A036015, A082303, A106507, A214263.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^4, x^4] / (QPochhammer[ -x^3, x^8] QPochhammer[ -x^5, x^8]), {x, 0, n}];
a[ n_] := SeriesCoefficient[ 1 / Product[ (1 + x^(8 k + 3)) (1 - x^(8 k + 4)) (1 + x^(8 k + 5)), {k, 0, Ceiling[ n/8]}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ 0, 0, 1, -1, 1, -1, 0, 0, 0, -1, 1, -1, 1, 0, 0, 0}[[Mod[k, 16, 1]]], {k, n}], {x, 0, n}];

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 0, 0, 0, 1, -1, 1, -1, 0, 0, 0, -1, 1, -1, 1, 0, 0][k%16 + 1]), n))};

Expansion of f(x^4, x^12) / f(x^3, x^5) where f(, ) is Ramanujan's general theta function.
Euler transform of period 16 sequence [ 0, 0, -1, 1, -1, 1, 0, 0, 0, 1, -1, 1, -1, 0, 0, 0, ...].
G.f.: (1 + x^4 + x^12 + x^24 + x^40 + ...) / (1 + x^3 + x^5 + x^14 + x^18 + ...). [Ramanujan]
G.f.: 1 - x^3 * (1 - x) / (1 - x^2) + x^8 * (1 - x) * (1 - x^3) / ((1 - x^2) * (1 - x^4)) - ... [Ramanujan]
a(n) = (-1)^n * A036015(n) = A029838(2*n + 1) = - A082303(2*n + 1).
Convolution product of A106507 and A214263.

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

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A036016 Number of partitions of n into parts not of form 4k+2, 8k, 8k+3 or 8k-3.

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A259774 Expansion of f(x, x^7) / f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.

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