A028939 -id:A028939 - cp's OEIS frontend

A007096 Expansion of theta_3 / theta_4.

1, 4, 8, 16, 32, 56, 96, 160, 256, 404, 624, 944, 1408, 2072, 3008, 4320, 6144, 8648, 12072, 16720, 22976, 31360, 42528, 57312, 76800, 102364, 135728, 179104, 235264, 307672, 400704, 519808, 671744, 864960, 1109904, 1419456, 1809568, 2299832

Offset: 0

N. J. A. Sloane, Simon Plouffe

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number of partitions of 2n into parts with 2 types c, c* of each part. The even parts appears with multiplicity 1 for each type. The odd parts appears with multiplicity 2 (cc or c*c* but not cc*, that is, no mixing is allowed). E.g., a(4)=8 because of 44*, 22*, 211, 21*1*, 2*1*1*, 2*11, 111*1*. - Noureddine Chair, Jan 27 2005
a(n) is the number of pairs of overpartitions into odd parts where the sum of all parts is equal to n. - Jeremy Lovejoy, Aug 29 2020

			G.f. = 1 + 4*q + 8*q^2 + 16*q^3 + 32*q^4 + 56*q^5 + 96*q^6 + 160*q^7 + 256*q^8 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Shane Chern, Dennis Eichhorn, Shishuo Fu, and James A. Sellers, Convolutive sequences, I: Through the lens of integer partition functions, arXiv:2507.10965 [math.CO], 2025. See pp. 4, 8, 13.
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. 11.
Bernard L.S. Lin, Arithmetic properties of overpartition pairs into odd parts, Electronic J. Combin. 19, 2012, Paper 17.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Self-convolution of A080054. - Vladeta Jovovic, Mar 22 2005
Cf. A014969, A001936, A001938, A079006, A127391, A127392.
Cf. A022577, A080054, A097243, A123655.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] / EllipticTheta[ 4, 0, q], {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m)^(-1/4), {q, 0, n}]]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[( QPochhammer[ -q, q^2] / QPochhammer[ q, q^2])^2, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ (Product[ 1 - (-q)^k, {k, n}] / Product[ 1 - q^k, {k, n}])^2, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
nmax=60; CoefficientList[Series[Product[((1+x^(2*k+1))/(1-x^(2*k+1)))^2, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 28 2015 *)

{a(n) = my(A, B); if( n<0, 0, A = 1 + 4*x; for( k=2, n, B = A + x^2 * O(x^k); A += Pol(2 * subst(B, x, x^2)^2 - B - 1/B) / x / 8); polcoeff(A, n))}; /* Michael Somos, Jul 07 2005*/

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)))^2, n))}; /* Michael Somos, Jan 01 2006 */

Euler transform of period 4 sequence [4, -2, 4, 0, ...]. - Vladeta Jovovic, Mar 22 2005
Expansion of eta(q^2)^6 /(eta(q)^4 * eta(q^4)^2) in powers of q.
Expansion of phi(q) / phi(-q) = chi(q)^2 / chi(-q)^2 = psi(q)^2 / psi(-q)^2 = phi(-q^2)^2 / phi(-q)^2 = phi(q)^2 / phi(-q^2)^2 = chi(-q^2)^2 / chi(-q)^4 = chi(q)^4 / chi(-q^2)^2 = f(q)^2 / f(-q)^2 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (1 - u^4) * (1 - v^4) - (1 - u*v)^4. - Michael Somos, Jan 01 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A028939.
Expansion of Jacobian elliptic function 1 / sqrt(k') in powers of q. - see Fine.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 1 + u^2 - 2*u*v^2. - Michael Somos, Jul 07 2005
Unique solution to f(x^2)^2 = (f(x) + 1 / f(x)) / 2 and f(0)=1, f'(0) nonzero.
G.f.: theta_3 / theta_4 = (Sum_{k} x^k^2) / (Sum_{k} (-x)^k^2) = (Product_{k>0} (1 - x^(4*k - 2)) / ((1 - x^(4*k - 1)) * (1 - x^(4*k - 3)))^2)^2.
A097243(n) = a(4*n). 8*A022577(n) = a(4*n + 2). a(n) = 4*A123655(n) if n>0. Convolution square of A080054.
Empirical: sum(exp(-Pi)^(n-1)*a(n),n=1..infinity) = 2^(1/4). - Simon Plouffe, Feb 20 2011
Empirical : sum(exp(-Pi*sqrt(2))^(n-1)*(-1)^(n+1)*a(n),n=1..infinity) = (-2+2*2^(1/2))^(1/4). - Simon Plouffe, Feb 20 2011
Empirical : sum(exp(-2*Pi)^(n-1)*a(n),n=1..infinity) = 1/2*(8+6*2^(1/2))^(1/4). - Simon Plouffe, Feb 20 2011
a(n) ~ exp(Pi*sqrt(n)) / (4*sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Aug 28 2015
G.f.: exp(4*Sum_{k>=1} sigma(2*k - 1)*x^(2*k-1)/(2*k - 1)). - Ilya Gutkovskiy, Apr 19 2019

A051138 Divisibility sequence associated with elliptic curve y^2 + y = x^3 - x and point (1, 0).

Original entry on oeis.org

0, 1, 1, -1, -5, -4, 29, 129, -65, -3689, -16264, 113689, 2382785, 7001471, -398035821, -7911171596, 43244638645, 6480598259201, 124106986093951, -5987117709349201, -541051130050800400, -4830209396684261199

Offset: 0

Michael Somos, Oct 12 1999

This is a strong divisibility sequence; that is, if n divides m, then a(n) divides a(m) and moreover for all positive integer n,m a(gcd(n, m)) = gcd(a(n), a(m)).
This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 1, y = -1, z = -5. - Michael Somos, Jul 07 2014
The elliptic curve y^2 + y = x^3 - x has LMFDB label 37.a1 (Cremona label 37a1). - Michael Somos, Feb 07 2024

			G.f. = x + x^2 - x^3 - 5*x^4 - 4*x^5 + 29*x^6 + 129*x^7 - 65*x^8 + ...

T. D. Noe, Table of n, a(n) for n = 0..100
Clark Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.
R. W. Gosper and Richard C. Schroeppel, Somos Sequence Near-Addition Formulas and Modular Theta Functions, arXiv:math/0703470 [math.NT], 2007.
LMFDB, Elliptic Curve 37.a1
Helmut Ruhland, Somos-4 and a quartic Surface in RP^3, arXiv:2312.02085 [math.AG], 2023.
Index to divisibility sequences

Cf. A006769, A006720, A028937, A028939.

a[n_ /; n < 0] := -a[-n]; a[0] = 0; A006769[n_] := (ClearAll[an]; an[] = 1; an[3] = -1; For[k = 5, k <= n, k++, an[k] = (an[k-1]*an[k-3] + an[k-2]^2)/an[k-4]]; an[n]); a[n] := A006769[2n]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Apr 11 2012, after 2nd formula *)

an = vector(200); an = concat([ 1, 1, -1, -5 ], an); for( n=5, length(an), an[ n ]=(an[ n-1 ] * an[ n-3 ] + an[ n-2 ]^2) / an[ n-4 ]); a(n) = an[ n ]

{a(n) = my(v = [1, 1, -1, -5]); if( n<0, -a(-n), if( n==0, 0, if( n<5, v[n], v = concat( v, vector(n - 4)); for( k=5, n, v[k] = (v[k-1] * v[k-3] + v[k-2]^2) / v[k-4]); v[n])))}; /* Michael Somos, Feb 12 2012 */

{a(n) = subst(elldivpol(ellinit([0, 0, 1, -1, 0]), n, 'x), 'x, 1)}; /* Michael Somos, Apr 21 2025 */

a(n) = (a(n-1) * a(n-3) + a(n-2)^2) / a(n-4).
a(n) = (-a(n-1) * a(n-4) + 5 * a(n-2) * a(n-3)) / a(n-5).
a(2*n + 1) = a(n+2) * a(n)^3 - a(n-1) * a(n+1)^3.
a(2*n) = a(n+2) * a(n) * a(n-1)^2 - a(n) * a(n-2) * a(n+1)^2.
a(-n) = -a(n). a(n) = A006769(2*n). a(n)^2 = A028937(n). |a(n)|^3 = A028939(n) for all n in Z.
0 = a(n)*a(n+4) - a(n+1)*a(n+3) - a(n+2)*a(n+2) for all n in Z. - Michael Somos, Jul 07 2014
0 = a(n)*a(n+5) + a(n+1)*a(n+4) - 5*a(n+2)*a(n+3) for all n in Z. - Michael Somos, Jul 07 2014

A028937 Denominator of x-coordinate of (2n)*P where P = (0,0) is the generator for rational points on the curve y^2 + y = x^3 - x.

Original entry on oeis.org

1, 1, 1, 25, 16, 841, 16641, 4225, 13608721, 264517696, 12925188721, 5677664356225, 49020596163841, 158432514799144041, 62586636021357187216, 1870098771536627436025, 41998153797159031581158401, 15402543997324146892198790401

Offset: 1

N. J. A. Sloane

			a(4) = 25 where 8P = (21/25, -69/125).

Seiichi Manyama, Table of n, a(n) for n = 1..106
LMFDB, Elliptic Curve 37.a1 (Cremona label 37a1)
B. Mazur, Arithmetic on curves, Bull. Amer. Math. Soc. 14 (1986), 207-259; see p. 225.

Cf. A006720, A051138, A028936 (numerator), A028938, A028939, A028941.

a(n)=denominator(ellmul(E,[0,0],2*n)[1]) \\ Charles R Greathouse IV, Mar 23 2022

P=(0, 0), 2P=(1, 0); if kP=(a, b) then (k+1)P = (a' = (b^2 - a^3)/a^2, b' = -1 - b*a'/a).
a(n) = A028941(2n). - Seiichi Manyama, Nov 19 2016
a(n) = a(-n) = b(n)*b(n+3) - b(n+1)*b(n+2) for all n in Z where b(n) = A006720(n). - Michael Somos, Mar 23 2022

A028936 Numerator of x-coordinate of (2n)*P where P is the generator for rational points on the curve y^2 + y = x^3 - x.

Original entry on oeis.org

1, 2, 6, 21, 161, 1357, 18526, 480106, 12551561, 683916417, 51678803961, 4881674119706, 997454379905326, 213822353304561757, 79799551268268089761, 53139223644814624290821, 36631192030206080565822006, 54202648602164057575419038802

Offset: 1

N. J. A. Sloane

			4P =(2, -3).
a(3) = 6 = 2*3 = A006720(4)*A006720(5). - _Michael Somos_, Apr 12 2020

Seiichi Manyama, Table of n, a(n) for n = 1..106
B. Mazur, Arithmetic on curves, Bull. Amer. Math. Soc. 14 (1986), 207-259; see p. 225.

Cf. A028937 (denominator), A028938, A028939, A028940.

Cf. A006720.

a(n)=numerator(ellmul(E,[0,0],2*n)[1]) \\ Charles R Greathouse IV, Mar 23 2022

P=(0, 0), 2P=(1, 0); if kP = (a, b) then (k+1)P = (a' = (b^2 - a^3)/a^2, b' = -1 - b*a'/a).
a(n) = A028940(2n). - Seiichi Manyama, Nov 19 2016
0 = a(n)*a(n+6) - 5*a(n+1)*a(n+5) + 4*a(n+2)*a(n+4) - 20*a(n+3)^2 for all n in Z. a(n) = A006720(n+1)*A006720(n+2). - Michael Somos, Apr 12 2020

cp's OEIS Frontend

A007096 Expansion of theta_3 / theta_4.

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A051138 Divisibility sequence associated with elliptic curve y^2 + y = x^3 - x and point (1, 0).

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A028937 Denominator of x-coordinate of (2n)*P where P = (0,0) is the generator for rational points on the curve y^2 + y = x^3 - x.

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A028936 Numerator of x-coordinate of (2n)*P where P is the generator for rational points on the curve y^2 + y = x^3 - x.

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A028938 Negative of numerator of y-coordinate of (2n)*P where P is generator for rational points on curve y^2 + y = x^3 - x.

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