A112610 -id:A112610 - cp's OEIS frontend

A207541 Expansion of phi(q)^3 * phi(-q) in powers of q where phi() is a Ramanujan theta function.

1, 4, 0, -16, -8, 24, 0, -32, 24, 52, 0, -48, -32, 56, 0, -96, 24, 72, 0, -80, -48, 128, 0, -96, 96, 124, 0, -160, -64, 120, 0, -128, 24, 192, 0, -192, -104, 152, 0, -224, 144, 168, 0, -176, -96, 312, 0, -192, 96, 228, 0, -288, -112, 216, 0, -288, 192, 320, 0

Offset: 0

Michael Somos, Feb 26 2012

sign

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

			1 + 4*q - 16*q^3 - 8*q^4 + 24*q^5 - 32*q^7 + 24*q^8 + 52*q^9 - 48*q^11 + ...

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. A096727, A097723, A112610, A121613, A139093, A208435.

a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, q]^3*EllipticTheta[3, 0, -q], {q, 0, n}]; Table[A207541[n], {n, 0, 50}] (* G. C. Greubel, Dec 16 2017 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^7 / (eta(x + A)^2 * eta(x^4 + A)^3))^2, n))}

Expansion of phi(-q^4)^4 + 4 * q * psi(-q^2)^4 = phi(q)^3 * phi(-q) = phi(q)^2 * phi(-q^2)^2 = psi(q)^4 * chi(-q^2)^6 = phi(-q^2)^6 / phi(-q)^2 = f(q)^6 / psi(q)^2 = f(q)^4 * chi(-q^2)^2 in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of (eta(q^2)^7 / (eta(q)^2 * eta(q^4)^3))^2 in powers of q.
Euler transform of period 4 sequence [ 4, -10, 4, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 128 (t/i)^2 g(t) where q = exp(2 Pi i t) and g(t) is g.f. for A112610.
G.f.: Product_{k>0} (1 - x^(2*k))^14 / ((1 - x^k)^4 * (1 - x^(4*k))^6).
a(3*n + 2) = 24 * A208435(n). a(4*n + 2) = 0. a(2*n + 1) = 4 * A121613(n). a(4*n) = A096727(n). a(4*n + 1) = 4 * A112610(n). a(4*n + 3) = -16 * A097723(n). Convolution square of A139093.

A321527 Expansion of x^3 * c(x^2) * c(x^4)^2 / (9 * c(x)) in powers of x where c() is a cubic AGM theta function.

Original entry on oeis.org

0, 0, 0, 1, -1, 0, 2, 0, -3, 4, 0, 0, 1, 0, 0, 6, -7, 0, 8, 0, -6, 8, 0, 0, -1, 0, 0, 13, -8, 0, 12, 0, -15, 12, 0, 0, 7, 0, 0, 14, -18, 0, 16, 0, -12, 24, 0, 0, -5, 0, 0, 18, -14, 0, 26, 0, -24, 20, 0, 0, 6, 0, 0, 32, -31, 0, 24, 0, -18, 24, 0, 0, 5, 0, 0, 31

Offset: 0

Michael Somos, Nov 12 2018

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Number 124 of the 126 eta-quotients listed in Table 1 of Williams 2012.

			G.f. = x^3 - x^4 + 2*x^6 - 3*x^8 + 4*x^9 + x^12 + 6*x^15 - 7*x^16 + ...

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..100000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
K. S. Williams, Fourier series of a class of eta quotients, Int. J. Number Theory 8 (2012), no. 4, 993-1004.

Cf. A008438, A033686, A097723, A112610, A144614, A224226, A229615, A321528.

A := Basis( ModularForms( Gamma0(12), 2), 76); A[4] - A[5];

a[ n_] := SeriesCoefficient[ x^3 QPochhammer[ x, x^2] QPochhammer[ x^12]^6 / (QPochhammer[ x^3, x^6]^3 QPochhammer[ x^4]^2), {x, 0, n}];
a[ n_] := With[ {s = If[ FractionalPart @ # > 0, 0, DivisorSigma[1, #]] &}, If[ n < 1, 0, s[n/3] - s[n/4] - s[n/6] + s[n/12]]];
a[ n_] := If[ n < 1, 0, Sum[ d {0, 0, 4, -3, 0, 2, 0, -3, 4, 0, 0, 0}[[Mod[d, 12, 1]]] / 12, {d, Divisors[n]}]];

{a(n) = if( n<1, 0, sumdiv( n, d, d * [0, 0, 0, 4, -3, 0, 2, 0, -3, 4, 0, 0][d%12 + 1] / 12))};

{a(n) = my(s = x -> if( frac(x), 0, sigma(x))); if( n<1, 0, s(n/3) - s(n/4) - s(n/6) + s(n/12))};

{a(n) = my(A); n-=3; if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^3 * eta(x^12 + A)^6 / (eta(x^2 + A) * eta(x^3 + A)^3 * eta(x^4 + A)^2), n))};

Expansion of x^3 * (psi(x^3) * psi(x^6))^3 / (psi(x) * psi(x^2)) in powers of x where psi() is a Ramanujan theta function.
Expansion of x^3 * chi(-x) * f(-x^12)^6 / (chi(-x^3)^3 * f(-x^4)^2) in powers of x where chi(), f() are Ramanujan theta functions.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (1/6) (t / i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A321528.
a(n) = s(n/3) - s(n/4) - s(n/6) + s(n/12) where s(x) = sum of divisors of x for integer x else 0.
a(6*n + 1) = a(6*n + 5) = a(12*n + 2) = a(12*n + 10) = 0.
a(n) = A224226(n) if n>0. a(2*n) = -A229615(n). a(6*n + 3) = A008438(n). a(12*n + 6) = 2*A008438(n).
a(12*n + 3) = A112610(n). a(12*n + 4) = -A144614(n). a(12*n + 8) = -3*A033686(n). a(12*n + 9) = 4*A097723(n).

A133657 Expansion of q * (phi(q) * psi(q^4))^2 in powers of q where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 4, 4, 0, 6, 16, 8, 0, 13, 24, 12, 0, 14, 32, 24, 0, 18, 52, 20, 0, 32, 48, 24, 0, 31, 56, 40, 0, 30, 96, 32, 0, 48, 72, 48, 0, 38, 80, 56, 0, 42, 128, 44, 0, 78, 96, 48, 0, 57, 124, 72, 0, 54, 160, 72, 0, 80, 120, 60, 0, 62, 128, 104, 0, 84, 192, 68, 0, 96

Offset: 1

Michael Somos, Sep 20 2007

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

			G.f. = q + 4*q^2 + 4*q^3 + 6*q^5 + 16*q^6 + 8*q^7 + 13*q^9 + 24*q^10 + ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A008438, A097723, A112610, A121455, A133690, A222068.

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, q^2]/2)^2, {q, 0, n}]; (* Michael Somos, Oct 30 2015 *)
a[n_] := Switch[IntegerExponent[n, 2], 0, DivisorSigma[1, n], 1, 4*DivisorSigma[1, n/2], , 0]; Array[a, 100] (* _Amiram Eldar, Nov 12 2022 *)

{a(n) = if( n<1, 0, if( n%2, sigma(n), if( n%4, 4 * sigma(n/2), 0)))};

{a(n) = my(A); if ( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 * eta(x^8 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^3) )^2, n))};

Expansion of (eta(q^2)^5 * eta(q^8)^2 / (eta(q)^2 * eta(q^4)^3))^2 in powers of q.
Euler transform of period 8 sequence [ 4, -6, 4, 0, 4, -6, 4, -4, ...].
a(n) is multiplicative with a(2) = 4, a(2^e) = 0 if e>1, a(p^e) = (p^(e+1) - 1) / (p - 1) if p>2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A133690.
a(4*n) = 0. a(4*n+2) = 4 * sigma(2*n+1). a(2*n+1) = sigma(2*n+1).
a(n) = -(-1)^n * A121455(n). Convolution square of A113411.
a(2*n + 1) = A008438. a(4*n + 1) = A112610(n). a(4*n + 3) = 4 * A097723(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/16 = 0.6168502... (A222068). - Amiram Eldar, Nov 12 2022

A252922 a(n) = sigma(n-1) + sigma(n-2) + sigma(n-3), with a(1)=0, a(2)=1, a(3)=4.

Original entry on oeis.org

0, 1, 4, 8, 14, 17, 25, 26, 35, 36, 46, 43, 58, 54, 66, 62, 79, 73, 88, 77, 101, 94, 110, 92, 120, 115, 133, 113, 138, 126, 158, 134, 167, 143, 165, 150, 193, 177, 189, 154, 206, 188, 228, 182, 224, 206, 234, 198, 244, 229, 274, 222, 263, 224, 272, 246, 312, 272, 290, 230, 318, 290, 326, 262, 327, 315, 355, 296

Offset: 1

Omar E. Pol, Dec 24 2014

This is also a rectangular array read by rows, with four columns, in which T(j,k) is the number of cells (also the area) of the j-th gap between the arms in the k-th quadrant of the spiral of the symmetric representation of sigma described in A239660, with j >= 1 and 1 <= k <= 4 and starting with T(1,1) = 0, see example.

We can find the spiral (mentioned above) on the terraces of the stepped pyramid described in A244050. - Omar E. Pol, Dec 07 2016

			a(5) = sigma(4) + sigma(3) + sigma(2) = 7 + 4 + 3 = 14. On the other hand a(5) = A024916(4) - A024916(1) = 15 - 1 = 14.
...
Also, if written as a rectangular array T(j,k) with four columns the sequence begins:
    0,   1,   4,   8;
   14,  17,  25,  26;
   35,  36,  46,  43;
   58,  54,  66,  62;
   79,  73,  88,  77;
  101,  94, 110,  92;
  120, 115, 133, 113;
  138, 126, 158, 134;
  167, 143, 165, 150;
  193, 177, 189, 154;
  206, 188, 228, 182;
  224, 206, 234, 198;
  244, 229, 274, 222;
  263, 224, 272, 246;
  312, 272, 290, 230;
  318, 290, 326, 262;
  ...
In this case T(2,1) = a(5) = 14.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000203, A010883, A024916, A092403, A112610, A193553, A196020, A236104, A237270, A237271, A237593, A239052, A239053, A239931-A239934, A239660, A240020, A244050, A245092, A262626.

L:= [0,0,0,seq(numtheory:-sigma(n), n=1..100)]:
L[1..101]+L[2..102]+L[3..103]; # Robert Israel, Dec 07 2016

a252922[n_] := Block[{f}, f[1] = 0; f[2] = 1; f[3] = 4;
  f[x_] := DivisorSigma[1, x - 1] + DivisorSigma[1, x - 2] +
DivisorSigma[1, x - 3]; Table[f[i], {i, n}]]; a252922[68] (* Michael De Vlieger, Dec 27 2014 *)

v=concat([0,1,4],vector(100,n,sigma(n)+sigma(n+1)+sigma(n+2))) \\ Derek Orr, Dec 30 2014

a(1) = 0, a(2) = sigma(1) = 1, a(3) = sigma(2) + sigma(1) = 4; for n >= 4, a(n) = sigma(n-1) + sigma(n-2) + sigma(n-3).

a(n) = A024916(n-1) - A024916(n-4) for n >= 5.

A133691 Expansion of (1 - (phi(-q) * phi(q^2))^2) / 4 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 4, -6, 6, -8, 8, -6, 13, -12, 12, -24, 14, -16, 24, -6, 18, -26, 20, -36, 32, -24, 24, -24, 31, -28, 40, -48, 30, -48, 32, -6, 48, -36, 48, -78, 38, -40, 56, -36, 42, -64, 44, -72, 78, -48, 48, -24, 57, -62, 72, -84, 54, -80, 72, -48, 80, -60, 60, -144

Offset: 1

Michael Somos, Sep 20 2007

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

			G.f. = q - 2*q^2 + 4*q^3 - 6*q^4 + 6*q^5 - 8*q^6 + 8*q^7 - 6*q^8 + 13*q^9 - ...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A000593, A008438, A046897, A097723, A111973, A112610, A133690.

a[ n_] := Which[ n < 1, 0, OddQ[n], DivisorSigma[ 1, n], True, -2 DivisorSum[ n/2, # Boole[Mod[#, 4] > 0] &]]; (* Michael Somos, Oct 30 2015 *)
a[ n_] := SeriesCoefficient[ (1 - (EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^2])^2) / 4, {q, 0, n}]; (* Michael Somos, Oct 30 2015 *)

{a(n) = if( n<1, 0, n%2, sigma(n), -2 * sumdiv(n/2, d, if(d%4, d)))};

Expansion of (1 - (eta(q)^2 * eta(q^4)^5 / (eta(q^2)^3 * eta(q^8)^2))^2) / 4 in powers of q.
a(n) is multiplicative with a(2) = -2, a(2^e) = -6 if e>1, a(p^e) = (p^(e+1) - 1) / (p - 1) if p>2.
a(n) = -4 * A133690(n) = -(-1)^n * A111973(n). a(2*n) = -2 * A046897(n). a(2*n + 1) = A008438(n). a(4*n) = -6 * A000593(n). a(4*n + 1) = A112610(n). a(4*n + 3) = 4 * A097723(n).
Dirichlet g.f.: zeta(s) * zeta(s-1) * (1 - 5/2^s + 1/2^(2*s-1) + 1/2^(3*s-3)). - Amiram Eldar, Oct 28 2023

A208451 Expansion of phi(q) * phi(-q)^3 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -4, 0, 16, -8, -24, 0, 32, 24, -52, 0, 48, -32, -56, 0, 96, 24, -72, 0, 80, -48, -128, 0, 96, 96, -124, 0, 160, -64, -120, 0, 128, 24, -192, 0, 192, -104, -152, 0, 224, 144, -168, 0, 176, -96, -312, 0, 192, 96, -228, 0, 288, -112, -216, 0, 288, 192, -320

Offset: 0

Michael Somos, Feb 26 2012

sign

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

			G.f. = 1 - 4*q + 16*q^3 - 8*q^4 - 24*q^5 + 32*q^7 + 24*q^8 - 52*q^9 + 48*q^11 + ...

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. A004011, A008438, A096727, A097723, A112610, A121613.

A := Basis( ModularForms( Gamma0(16), 2), 58); A[1] - 4*A[2] + 16*A[4] - 8*A[5]; /* Michael Somos, Aug 21 2014 */

a[ n_] := SeriesCoefficient[ (QPochhammer[ q]^2 QPochhammer[ q^2] / QPochhammer[ q^4])^2, {q, 0, n}]; (* Michael Somos, Aug 21 2014 *)

{a(n) = if( n<1, n==0, if( n%4 == 2, 0, -4 * if( n%2, (-1)^(n\2) * sigma(n), -2 * (-1)^(n\4) * sumdiv( n\4, d, if( d%4, d)))))};

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

Expansion of phi(-q^4)^4 - 4 * q * psi(-q^2)^4 = phi(q) * phi(-q)^3 = phi(-q)^2 * phi(-q^2)^2 = phi(-q^2)^6 / phi(q)^2 = psi(-q)^4 * chi(-q^2)^6 = f(-q)^4 * chi(-q^2)^2 = f(-q)^6 / psi(-q)^2 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of (eta(q)^2 * eta(x^2) / eta(x^4))^2 in powers of q.
Euler transform of period 4 sequence [ -4, -6, -4, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 5128 (t/i)^2 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A097723.
a(4*n + 2) = 0. a(2*n + 1) = -4 * A121613(n). a(4*n) = A096727(n). a(4*n + 1) = -4 * A112610(n). a(4*n + 3) = 16 * A097723(n). a(8*n) = A004011(n). a(8*n + 4) = -8 * A008438(n).

cp's OEIS Frontend

A207541 Expansion of phi(q)^3 * phi(-q) in powers of q where phi() is a Ramanujan theta function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A321527 Expansion of x^3 * c(x^2) * c(x^4)^2 / (9 * c(x)) in powers of x where c() is a cubic AGM theta function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

PARI

Formula

A133657 Expansion of q * (phi(q) * psi(q^4))^2 in powers of q where phi(), psi() are Ramanujan theta functions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A252922 a(n) = sigma(n-1) + sigma(n-2) + sigma(n-3), with a(1)=0, a(2)=1, a(3)=4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A133691 Expansion of (1 - (phi(-q) * phi(q^2))^2) / 4 in powers of q where phi() is a Ramanujan theta function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A208451 Expansion of phi(q) * phi(-q)^3 in powers of q where phi() is a Ramanujan theta function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula