A082303 -id:A082303 - cp's OEIS frontend

A001935 Number of partitions with no even part repeated; partitions of n in which no parts are multiples of 4.

1, 1, 2, 3, 4, 6, 9, 12, 16, 22, 29, 38, 50, 64, 82, 105, 132, 166, 208, 258, 320, 395, 484, 592, 722, 876, 1060, 1280, 1539, 1846, 2210, 2636, 3138, 3728, 4416, 5222, 6163, 7256, 8528, 10006, 11716, 13696, 15986, 18624, 21666, 25169, 29190, 33808, 39104, 45164

Offset: 0

N. J. A. Sloane, Simon Plouffe, Robert G. Wilson v

Also number of partitions of n where no part appears more than three times.
a(n) satisfies Euler's pentagonal number (A001318) theorem, unless n is in A062717 (see Fink et al.).
Also number of partitions of n in which the least part and the differences between consecutive parts is at most 3. Example: a(5)=6 because we have [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1] and [1,1,1,1,1]. - Emeric Deutsch, Apr 19 2006
Equals A000009 convolved with its aerated variant, = polcoeff A000009 * A000041 * A010054 (with alternate signs). - Gary W. Adamson, Mar 16 2010
Equals left border of triangle A174715. - Gary W. Adamson, Mar 27 2010
The Cayley reference is actually to A083365. - Michael Somos, Feb 24 2011
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution of A000009 and A035457. - Vaclav Kotesovec, Aug 23 2015
Convolution inverse is A082303. - Michael Somos, Sep 30 2017
The g.f. in the form Sum_{n >= 0} x^(n*(n+1)/2) * Product_{k = 1..n} (1+x^k)/(1-x^k) = Sum_{n >= 0} x^(n*(n+1)/2) * Product_{k = 1..n} (1+x^k)/(1+x^k-2*x^k) == Sum_{n >= 0} x^(n*(n+1)/2) (mod 2). It follows that a(n) is odd iff n = k*(k + 1)/2 for some nonnegative integer k. Cf. A333374. - Peter Bala, Jan 08 2025

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 9*x^6 + 12*x^7 + 16*x^8 + 22*x^9 + ...
G.f. = q + q^9 + 2*q^17 + 3*q^25 + 4*q^33 + 6*q^41 + 9*q^49 + 12*q^57 + 16*q^65 + 22*q^73 + ...
a(5)=6 because we have [5], [4,1], [3,2], [3,1,1], [2,1,1,1] and [1,1,1,1,1].

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.5.2).
M. D. Hirschhorn, The Power of q, Springer, 2017. See ped page 303ff.
R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 241.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
George E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573. (See Th. 9.)
George E. Andrews, Partition Identities for Two-Color Partitions, Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2021, Special Commemorative volume in honour of Srinivasa Ramanujan, 2021, 44, pp.74-80. hal-03498190. See Theorem 1.4 p. 75.
Riccardo Aragona, Roberto Civino, and Norberto Gavioli, A modular idealizer chain and unrefinability of partitions with repeated parts, arXiv:2301.06347 [math.RA], 2023.
Cristina Ballantine and Mircea Merca, Parity of sums of partition numbers and squares in arithmetic progressions, The Ramanujan Journal, 2016.
A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129.]
S.-C. Chen, On the number of partitions with distinct even parts, Discrete Math., 311 (2011), 940-943.
A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008).
M. D. Hirschhorn and J. A. Sellers, A Congruence Modulo 3 for Partitions into Distinct Non-Multiples of Four, Article 14.9.6, Journal of Integer Sequences, Vol. 17 (2014).
Joro, Is "OEIS A001935 Number of partitions with no even part repeated" efficiently computable mod 4?
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 15.
Mircea Merca, New relations for the number of partitions with distinct even parts, Journal of Number Theory 176 (July 2017), 1-12.
Alexander Patkowski, On some partitions where even parts do not repeat, Demonstratio Mathematica Volume 42, Issue 2 (Jun 2009), pp. 259-263.
Eric Weisstein's World of Mathematics, Partition Function b_k and Partition Function P.
Wikipedia, Glaisher's Theorem.

Cf. A000009, A000726, A001936, A035959, A035985, A042968, A061198, A061199, A070048, A081055, A081056, A083365, A098491, A098492, A219601.
Cf. A000041, A010054. - Gary W. Adamson, Mar 16 2010
Cf. A174715. - Gary W. Adamson, Mar 27 2010
Cf. A082303.
Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

a001935 = p a042968_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Sep 02 2012

g:=product((1+x^j)*(1+x^(2*j)),j=1..50): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=0..48); # Emeric Deutsch, Apr 19 2006
# second Maple program:
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
     `if`(irem(d, 4)=0, 0, d), d=divisors(j)), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Nov 24 2015

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q] / EllipticTheta[ 2, Pi/4, q^(1/2)] / (16 q)^(1/8), {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 4, n, 4}] / Product[ 1 - x^k, {k, n}], {x, 0, n}]; (* Michael Somos, Jul 08 2011 *)
CoefficientList[Series[Product[1+x^j+x^(2j)+x^(3j), {j,1,48}], {x,0,48}],x] (* Jean-François Alcover, May 26 2011, after Jon Perry *)
QP = QPochhammer; CoefficientList[QP[q^4]/QP[q] + O[q]^50, q] (* Jean-François Alcover, Nov 24 2015 *)
a[0] = 1; a[n_] := a[n] = Sum[a[n-j] DivisorSum[j, If[Divisible[#, 4], 0, #]&], {j, 1, n}]/n; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 19 2016, after Alois P. Heinz *)
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 4], 0, 2] ], {n, 0, 49}] (* Robert Price, Jul 28 2020 *)

{a(n) = if( n<0, 0, polcoeff( eta(x^4 + x * O(x^n)) / eta(x + x * O(x^n)), n))};

{a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint( 8*n + 1) - 1)\2, prod(i=1, k, (1 + x^i) / (x^-i - 1), 1 + x * O(x^n))), n))}; /* Michael Somos, Jun 01 2004 */

{a(n)=polcoeff(exp(sum(m=1, n+1, x^m/(1+(-x)^m+x*O(x^n))/m)),n)} \\ Paul D. Hanna, Jul 24 2013

Euler transform of period 4 sequence [ 1, 1, 1, 0, ...].
Expansion of q^(-1/8) * eta(q^4) / eta(q) in powers of q. - Michael Somos, Mar 19 2004
Expansion of psi(-x) / phi(-x) = psi(x) / phi(-x^2) = psi(x^2) / psi(-x) = chi(x) / chi(-x^2)^2 = 1 / (chi(x) * chi(-x)^2) = 1 / (chi(-x) * chi(-x^2)) = f(-x^4) / f(-x) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Jul 08 2011
G.f.: Product(j>=1, 1 + x^j + x^(2*j) + x^(3*j)). - Jon Perry, Mar 30 2004
G.f.: Product_{k>=1} (1+x^k)^(2-k%2). - Jon Perry, May 05 2005
G.f.: Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k-1)) = 1 + Sum_{k>0}(Product_{i=1..k} (x^i + 1) / (x^-i - 1)).
G.f.: Sum_{n>=0} ( x^(n*(n+1)/2) * Product_{k=1..n} (1+x^k)/(1-x^k) ). - Joerg Arndt, Apr 07 2011
G.f.: P(x^4)/P(x) where P(x) = Product_{k>=1} 1-x^k. - Joerg Arndt, Jun 21 2011
A083365(n) = (-1)^n a(n). Convolution square is A001936. a(n) = A098491(n) + A098492(n). a(2*n) = A081055(n). a(2*n + 1) = A081056(n).
G.f.: (1+ 1/G(0))/2, where G(k) = 1 - x^(2*k+1) - x^(2*k+1)/(1 + x^(2*k+2) + x^(2*k+2)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jul 03 2013
G.f.: exp( Sum_{n>=1} (x^n/n) / (1 + (-x)^n) ). - Paul D. Hanna, Jul 24 2013
a(n) ~ Pi * BesselI(1, sqrt(8*n + 1)*Pi/4) / (2*sqrt(8*n + 1)) ~ exp(Pi*sqrt(n/2)) / (4 * (2*n)^(3/4)) * (1 + (Pi/(16*sqrt(2)) - 3/(4*Pi*sqrt(2))) / sqrt(n) + (Pi^2/1024 - 15/(64*Pi^2) - 15/128) / n). - Vaclav Kotesovec, Aug 23 2015, extended Jan 14 2017
a(n) = (1/n)*Sum_{k=1..n} A046897(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
G.f. is a period 1 Fourier series which satisfies f(-1 / (256 t)) = 1/2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A082303. - Michael Somos, Sep 30 2017

More terms from James Sellers

A029838 Expansion of square root of q times normalized Hauptmodul for Gamma(4) in powers of q^8.

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, 229, 99, -256

Offset: 0

N. J. A. Sloane

sign

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

			G.f. = 1 + x - x^2 + x^4 - x^6 - x^7 + 2*x^8 + x^9 - 2*x^10 - x^11 + 2*x^12 + ...
G.f. = 1/q + q^7 - q^15 + q^31 - q^47 - q^55 + 2*q^63 + q^71 - 2*q^79 - q^87 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 221 Entry 1(i).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162; see Eqs. (9.1),(9.3).
J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann. 318 (2000), no. 2, 255-275.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions, q-Pochhammer Symbol

Cf. A029839, A082303, A083365.

a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^2] QPochhammer[ q^2, q^4], {q, 0, n}]; (* Michael Somos, Aug 20 2014 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -q, -q] / QPochhammer[ q^4, q^4], {q, 0, n}]; (* Michael Somos, Aug 20 2014 *)
a[ n_] := SeriesCoefficient[ q^(1/8) EllipticTheta[ 2, 0, q^(1/2)] / EllipticTheta[ 2, 0, q], {q, 0, n}]; (* Michael Somos, Aug 20 2014 *)
(QPochhammer[-x, x^2, 1/2] + O[x]^100)[[3]] (* Vladimir Reshetnikov, Nov 20 2016 *)

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

{a(n) = my(A); if( n<0, 0, A = contfracpnqn( matrix(2, (sqrtint(8*n + 1) + 1)\2, i, j, if( i==1, x^(j-1), 1 + if( j>1, x^(j-1))))); polcoeff(A[1,1] / A[2,1] + x * O(x^n), n))}; /* Michael Somos, Mar 02 2006 */

{a(n) = my(A, m); if( n<0, 0, A = 1 + O(x); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A2 = subst(A, x, x^2); A = sqrt((A2 + 2  * x / A2) / A)); polcoeff(A, n))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 / eta(x + A) / eta(x^4 + A)^2, n))};

Expansion of f(x) / f(-x^4) = phi(x) / psi(x) = psi(x) / psi(x^2) = phi(-x^2) / psi(-x) = chi(x) * chi(-x^2) = chi^2(x) * chi(-x) = chi^2(-x^2) / chi(-x) = (phi(x) / psi(x^2))^(1/2) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of q^(1/8) * eta(q^2)^3 / (eta(q) * eta(q^4)^2) in powers of q.
Euler transform of period 4 sequence [ 1, -2, 1, 0, ...].
Given g.f. A(x), then B(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = 4 + v^4 - u^4*v^2. - Michael Somos, Mar 02 2006
Given g.f. A(x), then B(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = u^4 - v^4 - 4*u*v + u^3*v^3. - Michael Somos, Mar 02 2006
Given g.f. A(x), then B(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = 2 + w^2 - u^2*v*w. - Michael Somos, Mar 02 2006
Given g.f. A(x), then B(q) = A(q^8) / q satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u2^2 + u6^2 - u1*u2*u3*u6. - Michael Somos, Mar 02 2006
G.f. A(x) satisfies A(x)^2 = (A(x^4) + 2*x / A(x^4)) / A(x^2). - Michael Somos, Mar 08 2004
G.f. A(x) satisfies A(x) = (A(x^2)^2+4*x/A(x^2)^2)^(1/4). - Joerg Arndt, Aug 06 2011
G.f.: Product_{k>0} (1 + x^(2*k - 1)) / (1 + x^(2*k)) = (Sum_{k>0} x^((k^2 - k)/2)) / (Sum_{k>0} x^(k^2 - k)).
G.f.: 1 + x / (1 + x + x^2 / (1 + x^2 + x^3 / (1 + x^3 + ...))).
A082303(n) = (-1)^n a(n). Convolution square is A029839. Convolution inverse is A083365.
G.f.: 2 - 2/(1+Q(0)), where Q(k)= 1 + x^(k+1) + x^(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 02 2013
G.f.: (-x; x^2){1/2}, where (a; q)_n is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 20 2016
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A109506(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 14 2017
abs(a(n)) ~ sqrt(sqrt(2) + (-1)^n) * exp(Pi*sqrt(n)/2^(3/2)) / (4*n^(3/4)). - Vaclav Kotesovec, Feb 07 2023

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

A258741 Expansion of f(x^3, x^5) / f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.

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

Offset: 0

Michael Somos, Nov 06 2015

sign

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

			G.f. = 1 - x + x^2 - x^3 + 2*x^4 - 2*x^5 + 2*x^6 - 3*x^7 + 4*x^8 - 5*x^9 + ...
G.f. = 1/q - q^15 + q^31 - q^47 + 2*q^63 - 2*q^79 + 2*q^95 - 3*q^111 + ...

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 16th 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, A036016, A082303, A106507, A214264.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^4, x^4] / (QPochhammer[ -x, x^8] QPochhammer[ -x^7, x^8]), {x, 0, n}];
a[ n_] := SeriesCoefficient[ 1 / Product[ (1 + x^(8 k + 1)) (1 - x^(8 k + 4)) (1 + x^(8 k + 7)), {k, 0, Ceiling[ n/8]}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{1, -1, 0, -1, 0, 0, 1, 0, 1, 0, 0, -1, 0, -1, 1, 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, 1, -1, 0, -1, 0, 0, 1, 0, 1, 0, 0, -1, 0, -1, 1][k%16 + 1]), n))};

Expansion of f(x^4, x^12) / f(x, x^7) where f(, ) is Ramanujan's general theta function.
Euler transform of period 16 sequence [ -1, 1, 0, 1, 0, 0, -1, 0, -1, 0, 0, 1, 0, 1, -1, 0, ...].
G.f.: 1 / (Product_{k>=0} (1 + x^(8*k + 1)) * (1 - x^(8*k + 4)) * (1 + x^(8*k + 7))).
G.f.: (1 + x^4 + x^12 + x^24 + x^40 + ...) / (1 + x + x^7 + x^10 + x^22 + ...). [Ramanujan]
G.f.: 1 - x * (1 - x) / (1 - x^2) + x^4 * (1 - x) * (1 - x^3) / ((1 - x^2) * (1 - x^4)) - ... [Ramanujan]
a(n) = (-1)^n * A036016(n) = A029838(2*n) = A082303(2*n).
Convolution product of A106507 and A214264.

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.

A318027 Expansion of Product_{k>=1} 1/((1 - x^k)(1 - x^(4k))).

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 13, 18, 29, 39, 57, 77, 112, 148, 205, 271, 372, 484, 647, 838, 1110, 1423, 1852, 2361, 3051, 3857, 4922, 6191, 7849, 9805, 12319, 15314, 19131, 23649, 29333, 36099, 44556, 54568, 66963, 81683, 99803, 121229, 147413, 178411, 216111, 260590, 314365, 377819, 454229

Offset: 0

Ilya Gutkovskiy, Aug 13 2018

nonn

Convolution of A000041 and A035444.
Convolution of A000712 and A082303.
Convolution inverse of A107034.
Number of partitions of n if there are 2 kinds of parts that are multiples of 4.

			a(5) = 8 because we have [5], [4, 1], [4', 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1] and [1, 1, 1, 1, 1].

Zakir Ahmed, Nayandeep Deka Baruah, Manosij Ghosh Dastidar, New congruences modulo 5 for the number of 2-color partitions, Journal of Number Theory, Volume 157, December 2015, Pages 184-198.
Index entries for sequences related to partitions

Cf. A000041, A000712, A002512 (self-convolution), A002513, A035444, A082303, A100853, A107034, A318026, A318028.

a:=series(mul(1/((1-x^k)*(1-x^(4*k))),k=1..55),x=0,49): seq(coeff(a,x,n),n=0..48); # Paolo P. Lava, Apr 02 2019

nmax = 48; CoefficientList[Series[Product[1/((1 - x^k) (1 - x^(4 k))), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[1/(QPochhammer[x] QPochhammer[x^4]), {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[Exp[Sum[x^k (1 + x^k + x^(2 k) + 2*x^(3 k))/(k (1 - x^(4 k))), {k, 1, nmax}]], {x, 0, nmax}], x]
Table[Sum[PartitionsP[k] PartitionsP[n - 4 k], {k, 0, n/4}], {n, 0, 48}]

G.f.: exp(Sum_{k>=1} x^k*(1 + x^k + x^(2*k) + 2*x^(3*k))/(k*(1 - x^(4*k)))).

a(n) ~ 5^(3/4) * exp(sqrt(5*n/6)*Pi) / (2^(13/4) * 3^(3/4) * n^(5/4)). - Vaclav Kotesovec, Aug 14 2018

A374080 Expansion of Product_{k>=1} (1 - x^(4k-1)) (1 - x^(4*k)).

Original entry on oeis.org

1, 0, 0, -1, -1, 0, 0, 0, -1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, -1, -1, 0, 0, 0, -1, 0, 1, 0, -1, -1, 0, 0, -1, -1, 1, 1, 0, -1, 0, 1, 0, -1, 0, 1, 1, -1, -1, 1, 1, 0, 0, 1, 2, -1, -1, 0, 1, 0, -2, 0, 2, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -2, 0, 1, -1, -2, 1, 2, 0, -2, -1, 2, 0, -2

Offset: 0

Ilya Gutkovskiy, Jun 27 2024

sign

Ilya Gutkovskiy, Scatterplot of a(n) up to n=10000

Cf. A010815, A014601, A035364, A081362, A082303, A106459, A137569, A274719, A284316, A301505, A374058, A374060, A374081.

nmax = 85; CoefficientList[Series[Product[(1 - x^(4 k - 1)) (1 - x^(4 k)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -(1/n) Sum[DivisorSum[k, # &, Or[Mod[#, 4] == 0, Mod[#, 4] == 3] &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 85}]

A374081 Expansion of Product_{k>=1} (1 - x^(4k-3)) (1 - x^(4*k)).

Original entry on oeis.org

1, -1, 0, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, 1, 0, -1, 0, 1, -1, 0, 1, 0, -1, 0, 0, 0, -1, 0, 2, -1, -1, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 1, 1, 0, -2, 1, 1, -2, 0, 2, 0, -1, 0, 1, 0, -1, 0, 2, -1, -2, 1, 1, -1, -1, 1, 2, -2, -1, 2, 0, -2, 0, 2, 0, -2, 0, 2, -1, -2, 1, 2, -1, -2, 1, 2

Offset: 0

Ilya Gutkovskiy, Jun 27 2024

sign

Ilya Gutkovskiy, Scatterplot of a(n) up to n=10000

Cf. A010815, A035362, A042948, A081362, A082303, A106459, A137569, A274719, A284313, A301504, A374058, A374060, A374080.

nmax = 85; CoefficientList[Series[Product[(1 - x^(4 k - 3)) (1 - x^(4 k)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -(1/n) Sum[DivisorSum[k, # &, Or[Mod[#, 4] == 0, Mod[#, 4] == 1] &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 85}]

A092924 Expansion of a Schwarzian ({f_{32|8}, tau} / (4*Pi)^2) in powers of q^8.

Original entry on oeis.org

1, -1008, 8304, -28224, 66672, -127008, 232512, -346752, 533616, -763056, 1046304, -1342656, 1866816, -2215584, 2856576, -3556224, 4269168, -4953312, 6286128, -6914880, 8400672, -9709056, 11060928, -12265344, 14941248, -15877008, 18252192, -20603520, 22935168, -24585120

Offset: 0

John McKay (mckay(AT)cs.concordia.ca), Apr 18 2004

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The q-series f_{32|8} is the g.f. for A082303. This is given on page 274 of McKay and Sebbar along with equation (8.1) which gives an expression for the g.f. A(q) of this sequence. - Michael Somos, Aug 15 2014

			G.f. = 1 - 1008*x + 8304*x^2 - 28224*x^3 + 66672*x^4 - 127008*x^5 + 232512*x^6 + ...
G.f. = 1 - 1008*q^8 + 8304*q^16 - 28224*q^24 + 66672*q^32 - 127008*q^40 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann., 318 (2000), 255-275.

Cf. A062248, A082303.

eta[q_]:= q^(1/24)*QPochhammer[q]; E4[0] := 1; E4[q_]:= 1 +240*Sum[k^3* q^k/(1 - q^k), {k, 1, 150}]; CoefficientList[Series[(21*E4[-q] - 16*E4[q^2])/5, {q, 0, 100}], q] (* G. C. Greubel, Jul 25 2018 *)

A = ModularForms( Gamma0(8), 4, prec=32) . basis(); A[1] - 1008*A[2] + 8304*A[3] + 66672*A[4]; # Michael Somos, Aug 15 2014

Expansion of (21 * E_4(-q) - 16 * E_4(q^2)) / 5 in powers of q. [McKay and Sebbar, equation (8.1)] - Michael Somos, Aug 15 2014

G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 16 (t/i)^4 f(t) where q = exp(2 Pi i t). - Michael Somos, Aug 15 2014

More terms from Michael Somos, Aug 15 2014

cp's OEIS Frontend

A001935 Number of partitions with no even part repeated; partitions of n in which no parts are multiples of 4.

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A029838 Expansion of square root of q times normalized Hauptmodul for Gamma(4) in powers of q^8.

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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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A258741 Expansion of f(x^3, x^5) / f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.

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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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A318027 Expansion of Product_{k>=1} 1/((1 - x^k)*(1 - x^(4*k))).

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A318027 Expansion of Product_{k>=1} 1/((1 - x^k)(1 - x^(4k))).

A374080 Expansion of Product_{k>=1} (1 - x^(4k-1)) (1 - x^(4*k)).

A374081 Expansion of Product_{k>=1} (1 - x^(4k-3)) (1 - x^(4*k)).