A006950 -id:A006950 - cp's OEIS frontend

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.

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)).

A059777 Number of self-conjugate three-quadrant Ferrers graphs that partition n.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 4, 6, 7, 9, 12, 16, 19, 24, 31, 39, 47, 58, 72, 89, 107, 129, 158, 192, 228, 273, 329, 393, 465, 551, 655, 776, 911, 1070, 1261, 1480, 1726, 2014, 2354, 2742, 3180, 3688, 4279, 4954, 5716, 6590, 7603, 8754, 10049, 11532

Offset: 0

N. J. A. Sloane, Feb 21 2001

nonn

G. E. Andrews, Three-quadrant Ferrers graphs, Indian J. Math., 42 (No. 1, 2000), 1-7.

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

Cf. A002129, A006950, A059776, A059778, A316675.

mul((1+q^(2*n+3))/(1-q^(2*n+2)), n=0..101); # g.f.

nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k + 1))/(1 - x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 26 2016 *)

G.f.: 1/((1+x)*Sum_{k>=0} (-x)^(k*(k+1)/2)). [Corrected by N. J. A. Sloane, Jul 10 2022 at the suggestion of Eduardo Brietzke.] a(n) = (1/n)*Sum_{k=1..n} (-1)^(k+1)*(A002129(k)-1)*a(n-k). A006950(n) = a(n-1) + a(n), n > 0. - Vladeta Jovovic, Sep 22 2002
G.f.: 1/((1+x)*G(0)), where G(k)= 1 - x^(2*k+1)/(1 - x^(2*k+2)/(x^(2*k+2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 20 2013
G.f.: conjecture: 1/(Q(0) - 1), where Q(k) = 1 + (-x)^k - (-x)^(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 25 2013
a(n) ~ exp(sqrt(n/2)*Pi)/(8*sqrt(2)*n). - Vaclav Kotesovec, Sep 26 2016
G.f.: Sum_{k>=0} x^(2*k) * Product_{j=1..k} (1+x^(2*j-1))/(1-x^(2*j)). - Seiichi Manyama, Jul 11 2018

A096981 Number of partitions of n into parts congruent to {0, 1, 3, 5} mod 6.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 6, 7, 10, 12, 15, 21, 25, 30, 39, 46, 56, 72, 85, 101, 125, 147, 175, 215, 252, 296, 356, 415, 487, 582, 676, 786, 927, 1072, 1244, 1460, 1682, 1939, 2255, 2588, 2976, 3446, 3942, 4510, 5189, 5916, 6751, 7739, 8797, 9999, 11406, 12927, 14657

Offset: 0

Noureddine Chair, Aug 19 2004

nonn

Also, number of partitions of n in which the distinct parts are prime to 3 and the unrestricted parts are multiples of 3.

The inverted graded parafermionic partition function. This g.f. is a generalization of A003105, A006950 and A096938

			a(11) = 15 because we can write 11 = 10+1 = 8+2+1 = 7+4 = 5+4+2 (parts do not contain multiple of 3) = 9+2 = 8+3 = 7+3+1 = 6+5 = 6+4+1 = 6+3+2 = 5+3+3 = 5+3+2+1 = 4+3+3+1 = 3+3+3+2.
1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 5*x^6 + 6*x^7 + 7*x^8 + 10*x^9 + ...
q^-5 + q^19 + q^43 + 2*q^67 + 2*q^91 + 3*q^115 + 5*q^139 + 6*q^163 + 7*q^187 + ...

T. M. Apostol, An Introduction to Analytic Number Theory, Springer-Verlag, NY, 1976

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.
Noureddine Chair, The Euler-Riemann Gases, and Partition Identities, arXiv:1306.5415 [math-ph], 23-June-2013.
Donald Spector, Duality, partial supersymmetry and arithmetic number theory, arXiv:hep-th/9710002, 1997.
Donald Spector, Duality, partial supersymmetry and arithmetic number theory, J. Math. Phys. Vol. 39, 1998, p. 1919.

Cf. A047273, A056970, A097451, A098884.

a096981 = p $ tail a047273_list where
   p _  0         = 1
   p ks'@(k:ks) m = if k > m then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Feb 19 2013

series(product(1/(1-x^k+x^(2*k)-x^(3*k)+x^(4*k)-x^(5*k)), k=1..150), x=0,100);

CoefficientList[ Series[ Product[ 1/(1 - x^k + x^(2k) - x^(3k) + x^(4k) - x^(5k)), {k, 55}], {x, 0, 53}], x] (* Robert G. Wilson v, Aug 21 2004 *)
nmax = 100; CoefficientList[Series[x^3*QPochhammer[-1/x^2, x^3] * QPochhammer[-1/x, x^3]/((1 + x)*(1 + x^2) * QPochhammer[x^3, x^3]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) / (eta(x + A) * eta(x^6 + A)), n))} /* Michael Somos, Jun 08 2012 */

Expansion of q^(5/24) * eta(q^2) / (eta(q) * eta(q^6)) in powers of q. - Michael Somos, Jun 08 2012
Euler transform of period 6 sequence [1, 0, 1, 0, 1, 1, ...]. - Vladeta Jovovic, Aug 20 2004
G.f.: 1/product_{k>=1}(1-x^k+x^(2*k)-x^(3*k)+x^(4*k)-x^(5*k)) = Product_{k>=1}(1+x^(3*k-1))(1+x^(3*k-2))/(1-x^(3*k)).
a(n) ~ exp(2*Pi*sqrt(n)/3) / (2*sqrt(6)*n). - Vaclav Kotesovec, Aug 31 2015

Better definition from Vladeta Jovovic, Aug 20 2004
More terms from Robert G. Wilson v, Aug 21 2004
Incorrect b-file replaced by Vaclav Kotesovec, Aug 31 2015

A273225 Number of bipartitions of n wherein odd parts are distinct (and even parts are unrestricted).

Original entry on oeis.org

1, 2, 3, 6, 11, 18, 28, 44, 69, 104, 152, 222, 323, 460, 645, 902, 1254, 1722, 2343, 3174, 4278, 5722, 7601, 10056, 13250, 17358, 22623, 29382, 38021, 48984, 62857, 80404, 102528, 130282, 165002, 208398, 262495, 329666, 412878, 515840

Offset: 0

M.S. Mahadeva Naika, May 18 2016

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number of bipartitions of 'n' wherein odd parts are distinct (and even parts are unrestricted).
G.f. is the square of the g.f. of A006950. - Vaclav Kotesovec, Mar 25 2017

			a(4)=11 because "(0,4)=(0,3+1)=(0,2+2)=(1,3)=(1,2+1)=(2,2)=(4,0)=(3+1,0)=(2+2,0)=(3,1)=(2+1,1)".
G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 11*x^4 + 18*x^5 + 28*x^6 + 44*x^7 + ... - _Michael Somos_, Mar 02 2019
G.f. = q^-1 + 2*q^3 + 3*q^7 + 6*q^11 + 11*q^15 + 18*q^19 + 28*q^23 + ... - _Michael Somos_, Mar 02 2019

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
M. D. Hirschhorn and J. A. Sellers, Arithmetic properties of partitions with odd parts distinct, Ramanujan J. 22 (2010), 273--284.
M. S. Mahadeva Naika and D. S. Gireesh, Arithmetic Properties of Partition k-tuples with Odd Parts Distinct, JIS, Vol. 19 (2016), Article 16.5.7
Michael Somos, Introduction to Ramanujan theta functions
L. Wang, Arithmetic properties of partition triples with odd parts distinct, Int. J. Number Theory, 11 (2015), 1791--1805.
L. Wang, Arithmetic properties of partition quadruples with odd parts distinct, Bull. Aust. Math. Soc., doi:10.1017/S0004972715000647.
L. Wang, New congruences for partitions where the odd parts are distinct, J. Integer Seq. (2015), article 15.4.2.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

For a version with signs see A274621.

Cf. A006950.

Digits:=200:with(PolynomialTools): with(qseries): with(ListTools):
GenFun:=series(etaq(q,2,100)^2/etaq(q,1,100)^2/etaq(q,4,100)^2,q,50):
CoefficientList(sort(convert(GenFun,polynom),q,ascending),q);

s = QPochhammer[-1, x]^2/(4*QPochhammer[x^4, x^4]^2) + O[x]^40; CoefficientList[s, x] (* Jean-François Alcover, May 20 2016 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2, x^4] / QPochhammer[ x])^2, {x, 0, n}]; (* Michael Somos, Mar 02 2019 *)

{a(n) = my(A); if( n<0, 0 , A = x * O(x^n); polcoeff( eta(x^2 + A)^2 / (eta(x + A) * eta(x^4 + A))^2, n))}; /* Michael Somos, Mar 02 2019 */

G.f.: Product_{k>=1} (1 + x^k)^2 / (1 - x^(4*k))^2, corrected by Vaclav Kotesovec, Mar 25 2017
Expansion of 1 / psi(-x)^2 in powers of x where psi() is a Ramanujan theta function.
a(n) ~ exp(Pi*sqrt(n))/(2^(5/2)*n^(5/4)). - Vaclav Kotesovec, Jul 05 2016
Euler transform of period 4 sequence [2, 0, 2, 2, ...]. - Michael Somos, Mar 02 2019

A296045 a(n) = [x^n] Product_{k>=1} ((1 + x^(2k-1))/(1 - x^(2k)))^n.

Original entry on oeis.org

1, 1, 3, 13, 55, 231, 981, 4222, 18351, 80320, 353453, 1562364, 6932185, 30856541, 137725710, 616190583, 2762605791, 12408541299, 55825435656, 251523510045, 1134741006825, 5125453110196, 23175983361270, 104899547541255, 475228898015025, 2154737528486881, 9777332125043577

Offset: 0

Ilya Gutkovskiy, Dec 03 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A006950, A106337, A192540, A273225, A273226, A273228, A295832, A296043, A296044.

Table[SeriesCoefficient[Product[((1 + x^(2 k - 1))/(1 - x^(2 k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[Product[((1 + x^k)/(1 - x^(4 k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[(2 (-x)^(1/8)/EllipticTheta[2, 0, Sqrt[-x]])^n, {x, 0, n}], {n, 0, 26}]
Table[(-1)^n * 2^n * SeriesCoefficient[1/(QPochhammer[-1, x]*QPochhammer[x^2])^n, {x, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 07 2020 *)
(* Calculation of constants {d,c}: *) Chop[{1/r, 4/Sqrt[Pi*(77/2 - 4*s*(-r*s)^(7/8) * Derivative[0, 0, 2][EllipticTheta][2, 0, Sqrt[-r*s]])]} /. FindRoot[{s == (2*(-r*s)^(1/8))/EllipticTheta[2, 0, Sqrt[-r*s]], 7*I*r + 2*(-r*s)^(7/8)*Sqrt[r*s] * Derivative[0, 0, 1][EllipticTheta][2, 0, Sqrt[-r*s]] == 0}, {r, 1/5}, {s, 2}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)

a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 - x^(4*k)))^n.

a(n) ~ c * d^n / sqrt(n), where d = 4.62579056836776492108784045382518984897... (see A192540) and c = 0.255113338880004277664416308115912337... - Vaclav Kotesovec, Dec 05 2017

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

Original entry on oeis.org

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

A340647 G.f.: Sum_{k>=0} x^(k^2) / Product_{j=1..k} (1 - x^(2*j))^2.

Original entry on oeis.org

1, 1, 0, 2, 1, 3, 2, 4, 5, 6, 8, 8, 14, 12, 20, 18, 31, 27, 42, 40, 60, 60, 80, 86, 111, 124, 146, 174, 199, 241, 262, 328, 353, 444, 464, 590, 620, 780, 812, 1020, 1075, 1326, 1400, 1710, 1833, 2198, 2370, 2804, 3072, 3570, 3936, 4522, 5048, 5713, 6414, 7190

Offset: 0

Vaclav Kotesovec, Jan 14 2021

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, Graph - the asymptotic ratio

Cf. A000700, A006950, A340623.

nmax = 100; CoefficientList[Series[Sum[x^(k^2)/Product[(1-x^(2*j))^2, {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

a(n) = A006950(n) - A340623(n).

a(n) ~ exp(Pi*sqrt(n/2)) / (4*sqrt(2)*n).

A192540 G.f.: A(x) = Series_Reversion(xG(x)) where G(x) = Sum_{n>=0} (-x)^(n(n+1)/2).

Original entry on oeis.org

1, 1, 2, 6, 20, 70, 255, 960, 3707, 14597, 58382, 236522, 968597, 4003061, 16674858, 69936760, 295092057, 1251747436, 5334958079, 22834290248, 98108081192, 422986894605, 1829443421394, 7935301625600, 34510975557383, 150456011512671, 657415433062780

Offset: 1

Paul D. Hanna, Jul 03 2011

nonn

Related q-series: Sum_{n>=0} (-q)^(n*(n+1)/2) = q^(-1/8)*eta(q)*eta(q^4)/eta(q^2) is a g.f. of A106459.

			G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 20*x^5 + 70*x^6 + 255*x^7 + ...
The g.f. A = A(x) satisfies the following relations:
(1) A = x/(1 - A - A^3 + A^6 + A^10 - A^15 - A^21 + A^28 + A^36 + ...).
(2) A = x/((1-A)*(1+A^2)* (1-A^2)*(1+A^4)* (1-A^3)*(1+A^6)* (1-A^4)*(1+A^8)*...).
(3) A = x/((1-A)*(1-A^4)* (1-A^3)*(1-A^8)* (1-A^5)*(1-A^12)* (1-A^7)*(1-A^16)*...).
(4) A = x*(1+A)/(1-A^2)* (1+A^3)/(1-A^4)* (1+A^5)/(1-A^6) * (1+A^7)/(1-A^8)*...
(5) A = x*(1-A^2)/(1-A)* (1-A^6)/(1-A^2)* (1-A^10)/(1-A^3)* (1-A^14)/(1-A^4)*...
(6) A = x*exp(A/(1-A) - A^2/(2*(1+A^2)) + A^3/(3*(1-A^3)) - A^4/(4*(1+A^4)) + ...).
(7) A = x*exp(A + A^2/2 + 4*A^3/3 + 5*A^4/4 + 6*A^5/5 +...+ A113184(n)*A^n/n + ...).

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000

Cf. A106459, A006950, A296045.

nmax:=27: with(gfun): f := proc(x): x*add((-x)^(n*(n+1)/2),n=0..nmax) end: S:=series(f(x),x,nmax): g:= seriestoseries(S,'revogf'): seq(coeftayl (g,x=0,n),n=1..nmax); # Johannes W. Meijer, Jul 04 2011

Rest[CoefficientList[InverseSeries[Series[x*EllipticTheta[2, 0, Sqrt[-x]] / (2*(-x)^(1/8)), {x, 0, 30}], x], x]] (* Vaclav Kotesovec, Aug 17 2015 *)
(* Calculation of constants {d,c}: *) Chop[{1/r, 8*(s/Sqrt[2*Pi*(77 - 8*(-s)^(7/8) *s*(Derivative[0, 0, 2][EllipticTheta][2, 0, Sqrt[-s]] / r))])} /. FindRoot[{2*r == -(-s)^(7/8)*EllipticTheta[2, 0, Sqrt[-s]], 2*(-s)^(11/8)*Derivative[0, 0, 1][EllipticTheta][2, 0, Sqrt[-s]] == 7*r}, {r, 1/5}, {s, 1/2}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)

{a(n)=polcoeff(serreverse(x*sum(m=0,sqrtint(2*n)+1,(-x)^(m*(m+1)/2)+x*O(x^n))),n)}

{a(n)=local(A=x+x^2);for(i=1,n,A=x/prod(m=1,n,(1 - A^m)*(1 + A^(2*m))+x*O(x^n)));polcoeff(A,n)}

{a(n)=local(A=x+x^2);for(i=1,n,A=x/prod(m=1,n\2,(1 - A^(2*m-1))*(1 - A^(4*m))+x*O(x^n)));polcoeff(A,n)}

{a(n)=local(A=x+x^2);for(i=1,n,A=x*prod(m=1,n\2,(1 + A^(2*m-1))/(1 - A^(2*m)+x*O(x^n))));polcoeff(A,n)}

{a(n)=local(A=x+x^2);for(i=1,n,A=x*prod(m=1,n,(1 - A^(4*m-2))/(1 - A^m+x*O(x^n))));polcoeff(A,n)}

{a(n)=local(A=x+x^2); for(i=1, n, A=x*exp(sum(m=1, n, -(-A+x*O(x^n))^m/(1+(-A)^m)/m))); polcoeff(A, n)}

{a(n)=if(n<1,0,(1/n)*polcoeff(x/prod(k=1,n,(1-x^k)*(1+x^(2*k)+x*O(x^n)))^n,n))}

{a(n)=local(A=x+x^2);for(i=1,n,A=x*exp(sum(m=1,n, A^m*sumdiv(m,d,(-1)^(m-d)*d)/m)+x*O(x^n)));polcoeff(A,n)}

G.f. satisfies:
(1) A(x) = x/[Sum_{n>=0} (-A(x))^(n*(n+1)/2)].
(2) A(x) = x/[Product_{n>=1} (1 - A(x)^n)*(1 + A(x)^(2*n))].
(3) A(x) = x/[Product_{n>=1} (1 - A(x)^(2*n-1))*(1 - A(x)^(4*n))].
(4) A(x) = x* Product_{n>=1} (1 + A(x)^(2*n-1))/(1 - A(x)^(2*n)).
(5) A(x) = x* Product_{n>=1} (1 - A(x)^(4*n-2))/(1 - A(x)^n).
(6) A(x) = x* exp( Sum_{n>=1} -(-A(x))^n/(n*(1 + (-A(x))^n)) ).
(7) A(x) = x* exp( Sum_{n>=1} A(x)^n*Sum_{d|n} (-1)^(n-d)*d/n ).
a(n) = [x^n] (1/n)*x/[Product_{k>=1} (1 - x^k)*(1 + x^(2*k))]^n for n >= 1.
a(n) ~ c * d^n / n^(3/2), where d = 4.6257905683677649210878404538251898489748116820946869227688637924996..., c = 0.1001072494040204029591345793571534412084516176488795... . - Vaclav Kotesovec, Aug 17 2015

A266462 The number of conjugacy classes of invertible n X n matrices over GF(2) which are squares of other such matrices.

Original entry on oeis.org

1, 1, 2, 5, 10, 20, 41, 82, 166, 334, 667, 1336, 2682, 5360, 10724, 21467, 42936, 85876, 171786, 343574, 687184, 1374427, 2748852, 5497766, 10995706, 21991402, 43982908, 87966150, 175932383, 351864964, 703730584, 1407461288, 2814923196, 5629847656, 11259695532

Offset: 0

Victor S. Miller, Dec 29 2015

nonn

It follows from the form of the generating function that a(n) is asymptotic to alpha*2^n where alpha = Product_{m>=1} (1-(1/16)^m)*(1-2*(1/4)^m)/((1-2*(1/16)^m)*(1-(1/4)^m)). [corrected by Jason Yuen, May 19 2025]

N. J. A. Sloane, Table of n, a(n) for n = 0..60 [Based on the table in Miller (2016)]
Victor S. Miller, Counting Matrices that are Squares, arXiv:1606.09299 [math.GR], 2016.
Index entries for matrices, binary, which are squares

Cf. A006950.

terms = 35; CoefficientList[Product[(1-2x^(2n))(1-x^(2n))/((1-2x^n) (1-2x^(4n))(1+x^(2n-1))), {n, 1, terms}] + O[x]^terms, x] (* Jean-François Alcover, Aug 06 2018 *)

G.f.: Product_{n>=1} (1-2*x^(2*n))*(1-x^(2*n))/((1-2*x^n)*(1-2*x^(4*n))*(1+x^(2*n-1))).

More terms from Alois P. Heinz, Dec 29 2015

A285458 Expansion of Product_{k>=1} ((1 + x^k) / (1 - x^(4*k)))^k.

Original entry on oeis.org

1, 1, 2, 5, 9, 17, 30, 54, 94, 161, 269, 449, 740, 1200, 1930, 3083, 4877, 7650, 11919, 18444, 28363, 43341, 65848, 99523, 149654, 223901, 333448, 494427, 729996, 1073408, 1572264, 2294389, 3336191, 4834261, 6981727, 10050944, 14424665, 20639641, 29447118

Offset: 0

Vaclav Kotesovec, Apr 19 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A015128, A000041, A285445, A006950.

Cf. A156616, A000219, A285446, A285459.

nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^(4*k)))^k, {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(1/12 + 3 * (13*Zeta(3))^(1/3) * n^(2/3) / 4) * (13*Zeta(3))^(7/36) / (2 * A * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962.

cp's OEIS Frontend

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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A059777 Number of self-conjugate three-quadrant Ferrers graphs that partition n.

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A096981 Number of partitions of n into parts congruent to {0, 1, 3, 5} mod 6.

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A273225 Number of bipartitions of n wherein odd parts are distinct (and even parts are unrestricted).

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A296045 a(n) = [x^n] Product_{k>=1} ((1 + x^(2*k-1))/(1 - x^(2*k)))^n.

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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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A340647 G.f.: Sum_{k>=0} x^(k^2) / Product_{j=1..k} (1 - x^(2*j))^2.

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A192540 G.f.: A(x) = Series_Reversion(x*G(x)) where G(x) = Sum_{n>=0} (-x)^(n*(n+1)/2).

A296045 a(n) = [x^n] Product_{k>=1} ((1 + x^(2k-1))/(1 - x^(2k)))^n.

A192540 G.f.: A(x) = Series_Reversion(xG(x)) where G(x) = Sum_{n>=0} (-x)^(n(n+1)/2).