A035162 -id:A035162 - cp's OEIS frontend

A128617 Expansion of q^2 * psi(q) * psi(q^15) in powers of q where psi() is a Ramanujan theta function.

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

Offset: 1

Michael Somos, Mar 13 2007

nonn

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

Also the number of positive odd solutions to equation x^2 + 15y^2 = 8n. - Seiichi Manyama, May 21 2017

			G.f. = x^2 + x^3 + x^5 + x^8 + x^12 + 2*x^17 + x^18 + x^20 + 2*x^23 + x^27 + x^30 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 377, Entry 9(i).

Seiichi Manyama, 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. A035162.

a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -60, #] - KroneckerSymbol[ 20, #] KroneckerSymbol[ -3, n/#] &] / 2]; (* Michael Somos, Nov 12 2015 *)
a[ n_] := SeriesCoefficient[ q^2 (QPochhammer[ q^2] QPochhammer[ q^30])^2 / (QPochhammer[ q] QPochhammer[ q^15]), {q, 0, n}]; (* Michael Somos, Nov 12 2015 *)

{a(n) = if( n<1, 0, sumdiv(n, d, kronecker(-60, d) - kronecker(20, d) * kronecker(-3, n/d) )/2)};

{a(n) = my(A); if( n<2, 0, n-=2; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^30 + A))^2 / (eta(x + A) * eta(x^15 + A)), n))};

Expansion of (eta(q^2) * eta(q^30))^2 / (eta(q) * eta(q^15)) in powers of q.
Euler transform of period 30 sequence [ 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 2, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -2, ...].
For n>0, n in A028955 equivalent to a(n) nonzero. If a(n) nonzero, a(n) = A082451(n) and a(n) = -A121362(n).
a(n)= (A082451(n) - A121362(n) )/2.
G.f.: x^2 * Product_{k>0} (1 - x^k) * (1 - x^(15*k)) * (1 + x^(2*k))^2 * (1 + x^(30*k))^2.

A033782 Product t2(q^d); d | 23, where t2 = theta2(q)/(2*q^(1/4)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

nonn

Also the number of positive odd solutions to equation x^2 + 23y^2 = 8(n + 3). - Seiichi Manyama, May 21 2017

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

Cf. A035162, A128617, A033790.

Expansion of q^(-3) * (eta(q^2) * eta(q^46))^2 / (eta(q) * eta(q^23)) in powers of q. - Seiichi Manyama, May 21 2017

More terms from Seiichi Manyama, May 21 2017

A033790 Product t2(q^d); d | 31, where t2 = theta2(q)/(2*q^(1/4)).

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0

Offset: 0

N. J. A. Sloane

nonn

Also the number of positive odd solutions to equation x^2 + 31y^2 = 8(n + 4). - Seiichi Manyama, May 21 2017

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

Cf. A035162, A128617, A033782.

Expansion of q^(-4) * (eta(q^2) * eta(q^62))^2 / (eta(q) * eta(q^31)) in powers of q. - Seiichi Manyama, May 21 2017

More terms from Seiichi Manyama, May 21 2017

A110399 Expansion of (theta_3(q)*theta_3(q^7) - 1)/2 in powers of q.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 2, 1, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 1, 2, 0, 0, 4, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 5, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0

Offset: 1

Michael Somos, Oct 22 2005

Half the number of integer solutions to x^2 + 7*y^2 = n. - Jianing Song, Nov 20 2019

			G.f. = x + x^4 + x^7 + 2*x^8 + x^9 + 2*x^11 + 3*x^16 + 2*x^23 + ...

Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 302, Entry 17(ii).

Cf. A033719 (number of integer solutions to x^2 + 7*y^2 = n).
Cf. A035162, A035182.
Similar sequences: A096936, A113406, A138806.

f[p_, e_] := If[MemberQ[{1, 2, 4}, Mod[p, 7]], e + 1, (1 + (-1)^e)/2]; f[2, e_] := e - 1; f[7, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 07 2023 *)

{a(n) = my(x); if( n<1, 0, x = valuation(n, 2); abs(x -1) * sumdiv(n/2^x, d, kronecker(-28, d)))};

{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, e-1,  p==7, 1, kronecker(-7, p)==-1, (1+(-1)^e)/2, e+1)))};

{a(n) = my(A); if( n<1, 0, A = x *O(x^n); polcoeff( (eta(x + A)^-2 * eta(x^2 + A)^5 * eta(x^4 + A)^-2 * eta(x^7 + A)^-2 * eta(x^14 + A)^5 * eta(x^28 + A)^-2 - 1)/2, n))};

a(n) is multiplicative with a(2^e) = |e-1|, a(7^e)= 1, a(p^e) = e+1 if p == 1, 2, 4 (mod 7), a(p^e) = (1+(-1)^e)/2 if p == 3, 5, 6 (mod 7).
G.f.: Sum_{k>0} Kronecker(-7, k) x^k/(1-(-x)^k).
G.f.: (theta_3(q)*theta_3(q^7) - 1)/2 where theta_3(q) = 1 + 2*(q + q^4 + q^9 + ...).
a(2*n + 1) = A035162(2*n + 1) = A035182(2*n + 1). A033719(n) = 2*a(n) if n > 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(7)) = 0.593705... . - Amiram Eldar, Nov 16 2023

A033766 Number of prime divisors (counted without multiplicity) of A138613(n).

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 11, 20, 29, 47, 78, 129, 230, 400, 710, 1221

Offset: 1

N. J. A. Sloane, Sep 20 2008

The old entry with this sequence number was a duplicate of A035162.

Cf. A138613, A079488, A001221.

PrimeNu[NestList[DivisorSigma[2,#]&,2,12]] (* Harvey P. Dale, Aug 19 2017 *)

a(10)-a(12) from Donovan Johnson, Nov 17 2008

a(13)-a(16) from Daniel Suteu, Dec 11 2019

A033806 Product t2(q^d); d | 47, where t2 = theta2(q)/(2*q^(1/4)).

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

nonn

Also the number of positive odd solutions to equation x^2 + 47y^2 = 8(n + 6). - Seiichi Manyama, May 27 2017

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

Cf. A035162, A128617, A033782, A033790.

Expansion of q^(-6) * (eta(q^2) * eta(q^94))^2 / (eta(q) * eta(q^47)) in powers of q. - Seiichi Manyama, May 27 2017

More terms from Seiichi Manyama, May 27 2017

A121454 Expansion of q * psi(-q) * psi(-q^7) in powers of q where psi(q) is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Jul 30 2006

Ramanujan theta functions: f(q) := Product_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Product_{k>=0} (1+q^(2k+1)) (A000700).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A035162.

eta[q_]:= q^(1/24)*QPochhammer[q]; Rest[CoefficientList[Series[eta[q] eta[q^4] eta[q^7] eta[q^28]/(eta[q^2] eta[q^14]), {q, 0, 100}], q]] (* G. C. Greubel, Apr 19 2018 *)

{a(n)=if(n<1, 0, -(-1)^n*sumdiv(n,d,kronecker(-28,d)))}

{a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x+A)*eta(x^4+A)*eta(x^7+A)*eta(x^28+A)/eta(x^2+A)/eta(x^14+A), n))}

Expansion of eta(q)*eta(q^4)*eta(q^7)*eta(q^28)/(eta(q^2)*eta(q^14)) in powers of q.
Euler transform of period 28 sequence [ -1, 0, -1, -1, -1, 0, -2, -1, -1, 0, -1, -1, -1, 0, -1, -1, -1, 0, -1, -1, -2, 0, -1, -1, -1, 0, -1, -2, ...].
Moebius transform is period 28 sequence [ 1, -2, -1, 0, -1, 2, 0, 0, 1, 2, 1, 0, -1, 0, 1, 0, -1, -2, -1, 0, 0, -2, 1, 0, 1, 2, -1, 0, ...].
Multiplicative with a(2^e) = -1 if e>0, a(7^e) = 1, a(p^e) = e+1 if p == 1, 2, 4 (mod 7), a(p^e) = (1+(-1)^e)/2 if p == 3, 5, 6 (mod 7).
G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)= u^4*w*v +2*u^3*w*v^2 +2*u^2*w^2*v^2 +4*u^3*w^3 +4*u^3*w^2*v +8*u*w^4*v +8*u*w^3*v^2 +8*u^2*w^3*v -u^2*v^4 -2*u^2*w*v^3 -4*w^2*v^4 -4*u*w^2*v^3.
G.f.: Sum_{k>0} (-1)^k *x^k*(1-x^(2*k))*(1-x^(4*k))*(1-x^(6*k))/(1-x^(14*k)) = x * Product_{k>0} (1-x^k)*(1+x^(2*k))*(1-x^(7*k))*(1+ x^(14*k)).
a(7*n) = a(n). a(7*n+3) = a(7*n+5) = a(7*n+6) = 0.
a(n) = (-1)^(n+1)*A035162(n).

A133827 Number of solutions to x + 7 * y = 2 * n in triangular numbers.

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 0, 0, 0, 0, 0, 2, 1, 0, 2, 0, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 2, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 2, 0, 2, 0, 0, 0, 3, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0

Offset: 0

Michael Somos, Sep 25 2007, Oct 04 2008

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

G.f. is called omega(q) by Berkovich and Yesilyurt.

			G.f. = 1 + x^3 + x^4 + 2*x^5 + 2*x^11 + x^12 + 2*x^14 + 2*x^18 + 2*x^21 + x^24 + ...
G.f. = q + q^7 + q^9 + 2*q^11 + 2*q^23 + q^25 + 2*q^29 + 2*q^37 + 2*q^43 + q^49 + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Alexander Berkovich and Hamza Yesilyurt, New Identities For 7-cores with Prescribed BG-Rank, arXiv:math/0603150 [math.NT], 2006-2007, page 3, equation (1.19).
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A002652, A033719, A035162, A035182, A110399, A121454.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, Mod[#, 2] KroneckerSymbol[ -28, #] &]]; (* Michael Somos, Oct 30 2015 *)
a[ n_] := SeriesCoefficient[ (1/4) EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^(7/2)], {x, 0, 2 n + 1}]; (* Michael Somos, Oct 30 2015 *)

{a(n) = if( n<0, 0, n = 2*n + 1; sumdiv(n, d, (d%2) * kronecker( -28, d)))};

{a(n) = my(A, p, e); if( n<0, 0, n = 2*n + 1; A = factor(n); prod(k = 1, matsize(A)[1], [p, e] = A[k, ]; if(p == 2, 0, p == 7, 1, 1 == kronecker( -7, p), e + 1, 1-e%2)))};

Expansion of psi(x^4) * phi(x^14) + x^3 * psi(x^28) * phi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions.
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(7^e) = 1, b(p^e) = (1 + (-1)^e) / 2 if p == 3, 5, 6 (mod 7), b(p^e) = e + 1 if p == 1, 2, 4 (mod 7).
a(7*n + 1) = a(7*n + 2) = a(7*n + 6) = 0. a(7*n + 3) = a(n).
Expansion of psi(q) * psi(q^7) - q * psi(q^2) * psi(q^14) = (psi(q) * psi(q^7) + psi(-q) * psi(-q^7)) / 2 in powers of q^2 where psi() is a Ramanujan theta function.
a(n) = A035162(2*n + 1) = A035182(2*n + 1) = A110399(2*n + 1) = A121454(2*n + 1).
2 * a(n) = A002652(2*n + 1) = A033719(2*n + 1). - Michael Somos, Dec 30 2016
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(7)) = 0.593705... . - Amiram Eldar, Dec 29 2023

A286813 Number of positive odd solutions to equation x^2 + 8y^2 = 8n + 9.

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, May 28 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Simon Plouffe, Conjectures of the OEIS, as of June 20, 2018.

Related to the number of positive odd solutions to equation x^2 + k*y^2 = 8*n + k + 1: A008441 (k=1), A033761 (k=2), A033762 (k=3), A053692 (k=4), A033764 (k=5), A259896 (k=6), A035162 (k=7), this sequence (k=8).

Expansion of q^(-9/8) * (eta(q^2) * eta(q^16))^2 / (eta(q) * eta(q^8)) in powers of q.

Euler Transform of -(-2*x^8-x^7-1)/(x^9+x^8+x+1) (o.g.f.). - Simon Plouffe, Jun 23 2018

A287619 Number of positive odd solutions to equation x^2 + 39y^2 = 8*(n + 5).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, May 28 2017

nonn

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

Cf. A035162, A128617.

Number of positive odd solutions to equation x^2 + (8*k - 1)*y^2 = 8*(n + k): A033782 (k=3), A033790 (k=4), this sequence (k=5), A033806 (k=6).

Expansion of q^(-5) * (eta(q^2) * eta(q^78))^2 / (eta(q) * eta(q^39)) in powers of q.

cp's OEIS Frontend

A128617 Expansion of q^2 * psi(q) * psi(q^15) in powers of q where psi() is a Ramanujan theta function.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A033782 Product t2(q^d); d | 23, where t2 = theta2(q)/(2*q^(1/4)).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A033790 Product t2(q^d); d | 31, where t2 = theta2(q)/(2*q^(1/4)).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A110399 Expansion of (theta_3(q)*theta_3(q^7) - 1)/2 in powers of q.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A033766 Number of prime divisors (counted without multiplicity) of A138613(n).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A033806 Product t2(q^d); d | 47, where t2 = theta2(q)/(2*q^(1/4)).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A121454 Expansion of q * psi(-q) * psi(-q^7) in powers of q where psi(q) is a Ramanujan theta function.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A133827 Number of solutions to x + 7 * y = 2 * n in triangular numbers.

Views

Author

A286813 Number of positive odd solutions to equation x^2 + 8y^2 = 8n + 9.