A057538 -id:A057538 - cp's OEIS frontend

A093722 Integers of the form (k^2 - 1) / 120.

0, 1, 3, 7, 8, 14, 20, 29, 31, 42, 52, 66, 69, 85, 99, 118, 122, 143, 161, 185, 190, 216, 238, 267, 273, 304, 330, 364, 371, 407, 437, 476, 484, 525, 559, 603, 612, 658, 696, 745, 755, 806, 848, 902, 913, 969, 1015, 1074, 1086, 1147, 1197, 1261, 1274, 1340

Offset: 1

Michael Somos, Apr 13 2004

This is "one-fifteenth of triangular numbers (integers only)". - Vladimir Joseph Stephan Orlovsky, Mar 04 2009

The sequence terms are the exponents in the expansion of Product_{n >= 1} (1 - q^n)/( (1 - q^(10*n-2))*(1 - q^(10*n-8)) ) = 1 - q - q^3 + q^7 + q^8 - q^14 - q^20 + + - - ... . - Peter Bala, Dec 26 2024

Harry J. Smith, Table of n, a(n) for n = 1..20000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-2,0,0,-1,1).

Cf. A002381, A057538, A113430, A214429, A379210.

A093722 := proc(q) local n;
for n from 0 to q do
 if type(sqrt(120*n+1), integer) then print(n);
fi; od; end:
A093722(1500); # Peter Bala, Dec 26 2024

Select[Table[(n^2-1)/120,{n,0,700}],IntegerQ] (* Harvey P. Dale, Nov 26 2010 *)

{a(n) = (((n\4 * 3 + n%4) * 10 + (-1)^(n\2))^2 - 1) / 120 } /* Michael Somos, Oct 17 2006 */

|A113430(n-1)| is the characteristic function of the numbers in A093722.
a(-1 - n) = a(n). a(n) = (A057538(n) * 2 - 1) / 120.
G.f.: -x^2*(1+2*x+4*x^2+x^3+4*x^4+x^6+2*x^5) / ( (1+x)^2*(x^2+1)^2*(x-1)^3 ). - R. J. Mathar, Jun 09 2013
From Peter Bala, Dec 26 2024: (Start)
a(n) is quasi-polynomial in n
a(4*n) = n*(15*n + 1)/2; a(4*n+1) = (3*n + 1)*(5*n + 2)/2;
a(4*n+2) = (3*n + 2)*(5*n + 3)/2; a(4*n+3) = (n + 1)*(15*n + 14)/2.
For 0 <= k <= 3, a(4*n+k) = (N_k(n)^2 - 1)/120, where N_0(n) = 30*n + 1, N_1(n) = 30*n + 11, N_2(n) = 30*n + 19 and N_3(n) = 30*n + 29. (End)

More terms from Harvey P. Dale, Nov 26 2010

Offset corrected to 1 by Ray Chandler, Jul 29 2019

A057539 Birthday set of order 7, i.e., numbers congruent to +- 1 modulo 2, 3, 4, 5, 6 and 7.

Original entry on oeis.org

1, 29, 41, 71, 139, 169, 181, 209, 211, 239, 251, 281, 349, 379, 391, 419, 421, 449, 461, 491, 559, 589, 601, 629, 631, 659, 671, 701, 769, 799, 811, 839, 841, 869, 881, 911, 979, 1009, 1021, 1049, 1051, 1079, 1091, 1121, 1189, 1219, 1231, 1259, 1261, 1289

Offset: 1

Andrew R. Feist (andrewf(AT)math.duke.edu), Sep 06 2000

Integers of the form sqrt(840*k+1) for k >= 0. - Boyd Blundell, Jul 10 2021

Ray Chandler, Table of n, a(n) for n = 1..10000
A. Feist, On the Density of Birthday Sets, The Pentagon, 60 (No. 1, Fall 2000), 31-35.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,1,-1).

Cf. A057538, A057540, A057541.

LinearRecurrence[{1,0,0,0,0,0,0,1,-1},{1,29,41,71,139,169,181,209,211},50] (* Harvey P. Dale, Sep 24 2014 *)

is_A057539(n,m=[2,3,4,5,6,7])=!for(i=1,#m,abs((n+1)%m[i]-1)==1||return)

is(n)=for(i=4,7,if(abs(centerlift(Mod(n,i)))!=1, return(0))); 1 \\ Charles R Greathouse IV, Oct 20 2014

def ok(n): return all(n%d in [1, d-1] for d in range(2, 8))
def aupto(nn): return [m for m in range(1, nn+1) if ok(m)]
print(aupto(1300)) # Michael S. Branicky, Jan 29 2021

G.f.: x*(1 + 28*x + 12*x^2 + 30*x^3 + 68*x^4 + 30*x^5 + 12*x^6 + 28*x^7 + x^8) / ((1+x)*(x^2+1)*(x^4+1)*(x-1)^2). - R. J. Mathar, Oct 08 2011
a(n) = a(n-8) + 210 = a(n-1) + a(n-8) - a(n-9). - Charles R Greathouse IV, Oct 20 2014
a(n) = 105n/4 + O(1). - Charles R Greathouse IV, Oct 20 2014

Offset corrected to 1 by Ray Chandler, Jul 29 2019

A057540 Birthday set of order 8: i.e., numbers congruent to +- 1 modulo 2, 3, 4, 5, 6, 7 and 8.

Original entry on oeis.org

1, 41, 71, 169, 209, 239, 281, 391, 449, 559, 601, 631, 671, 769, 799, 839, 841, 881, 911, 1009, 1049, 1079, 1121, 1231, 1289, 1399, 1441, 1471, 1511, 1609, 1639, 1679, 1681, 1721, 1751, 1849, 1889, 1919, 1961, 2071, 2129, 2239, 2281, 2311, 2351, 2449

Offset: 1

Andrew R. Feist (andrewf(AT)math.duke.edu), Sep 06 2000

			2129 is on the list because it is congruent to 1 mod 2, -1 mod 3, 1 mod 4, -1 mod 5, -1 mod 6, 1 mod 7 and 1 mod 8.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Colin Barker)
A. Feist, On the Density of Birthday Sets, The Pentagon, 60 (No. 1, Fall 2000), 31-35.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A007310, A057538, A057539 and A057541 are also birthday sets.

bso8Q[n_]:=Module[{s1=Mod[n,Range[2,8]],s2},s2=Abs[s1-Range[2,8]];AllTrue[ Thread[{s1,s2}],MemberQ[#,1]&]]; Select[Range[2500],bso8Q] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 18 2021 *)

Vec(x*(x^16 +40*x^15 +30*x^14 +98*x^13 +40*x^12 +30*x^11 +42*x^10 +110*x^9 +58*x^8 +110*x^7 +42*x^6 +30*x^5 +40*x^4 +98*x^3 +30*x^2 +40*x +1) / ((x -1)^2*(x +1)*(x^2 +1)*(x^4 +1)*(x^8 +1)) + O(x^100)) \\ Colin Barker, Mar 16 2015

G.f.: x*(x^16 +40*x^15 +30*x^14 +98*x^13 +40*x^12 +30*x^11 +42*x^10 +110*x^9 +58*x^8 +110*x^7 +42*x^6 +30*x^5 +40*x^4 +98*x^3 +30*x^2 +40*x +1) / ((x -1)^2*(x +1)*(x^2 +1)*(x^4 +1)*(x^8 +1)). - Colin Barker, Mar 16 2015

Offset corrected to 1 by Ray Chandler, Jul 29 2019

A057541 Birthday set of order 9: i.e., numbers congruent to +- 1 modulo 2, 3, 4, 5, 6, 7, 8 and 9.

Original entry on oeis.org

1, 71, 449, 559, 631, 881, 1009, 1079, 1441, 1511, 1639, 1889, 1961, 2071, 2449, 2519, 2521, 2591, 2969, 3079, 3151, 3401, 3529, 3599, 3961, 4031, 4159, 4409, 4481, 4591, 4969, 5039, 5041, 5111, 5489, 5599, 5671, 5921, 6049, 6119, 6481, 6551, 6679

Offset: 1

Andrew R. Feist (andrewf(AT)math.duke.edu), Sep 06 2000

			5599 is on the list because it is congruent to 1 mod 2, 1 mod 3, -1 mod 4, -1 mod 5, 1 mod 6, -1 mod 7, -1 mod 8 and 1 mod 9.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Colin Barker)
A. Feist, On the Density of Birthday Sets, The Pentagon, 60 (No. 1, Fall 2000), 31-35.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A007310, A057538, A057539 and A057540 are other birthday sets.

LinearRecurrence[{1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1},{1,71,449,559,631,881,1009,1079,1441,1511,1639,1889,1961,2071,2449,2519,2521},80] (* Harvey P. Dale, Feb 20 2022 *)

Vec(x*(x^16 +70*x^15 +378*x^14 +110*x^13 +72*x^12 +250*x^11 +128*x^10 +70*x^9 +362*x^8 +70*x^7 +128*x^6 +250*x^5 +72*x^4 +110*x^3 +378*x^2 +70*x +1) / ((x -1)^2*(x +1)*(x^2 +1)*(x^4 +1)*(x^8 +1)) + O(x^100)) \\ Colin Barker, Mar 16 2015

G.f.: x*(x^16 +70*x^15 +378*x^14 +110*x^13 +72*x^12 +250*x^11 +128*x^10 +70*x^9 +362*x^8 +70*x^7 +128*x^6 +250*x^5 +72*x^4 +110*x^3 +378*x^2 +70*x +1) / ((x -1)^2*(x +1)*(x^2 +1)*(x^4 +1)*(x^8 +1)). - Colin Barker, Mar 16 2015

Offset corrected to 1 by Ray Chandler, Jul 29 2019

A113430 Expansion of f(-x, -x^2) * f(-x^10, -x^20) / f(-x^2, -x^8) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Oct 31 2005

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

This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = x^4, b = x.

			G.f. = 1 - x - x^3 + x^7 + x^8 - x^14 - x^20 + x^29 + x^31 - x^42 - x^52 + ...
G.f. = q - q^121 - q^361 + q^841 + q^961 - q^1681 - q^2401 + q^3481 + q^3721 + ...

George E. Andrews, Richard Askey and Ranjan Roy, Special Functions, Cambridge University Press, 1999.

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
Eric Weisstein's World of Mathematics, Quintuple Product Identity

Cf. A003114, A010815, A057538, A093722.

a[ n_] := SeriesCoefficient[ QPochhammer[ x] / (QPochhammer[ x^2, x^10] QPochhammer[ x^8, x^10]), {x, 0, n}]; (* Michael Somos, Jan 06 2016 *)

{a(n) = my(m); if( n<0 || !issquare( n*120 + 1, &m) || 1!=gcd(m, 30), 0, (-1)^(m%30\10))};

{a(n) = if( n<0, 0, polcoeff( prod( k=1, n, 1 - x^k * [1, 1, 0, 1, 1, 1, 1, 1, 0, 1][k%10 + 1], 1 + x * O(x^n)), n))};

Expansion of f(x^7, x^8) - x * f(x^2, x^13) in power of x.
Expansion of G(x^2) * f(-x) where G() is the g.f. of A003114.
Euler transform of period 10 sequence [ -1, 0, -1, -1, -1, -1, -1, 0, -1, -1, ...].
|a(n)| is the characteristic function of the numbers in A093722.
The exponents in the q-series q * A(q^120) are the square of the numbers in A057538.
G.f.: Prod_{k>0} (1 - x^k) / ((1 - x^(10*k - 2)) * (1 - x^(10*k - 8))) = Sum_{k in Z} x^((15*k^2 + k) / 2) - x^((15*k^2 - 11*k + 2) / 2).
A(q^2) = 1 + Sum_{n >= 0} q^(n^2) * Product_{k >= 2*n+1} 1 - q^k = 1 - q^2 - q^6 + q^14 + q^16 - q^28 - q^40  +  + - - . See Andrews et al., p. 591, Exercise 6(a). - Peter Bala, Dec 22 2024

A113681 Expansion of f(-x^2, -x^3)^2 / f(-x, -x^2) in powers of x where f() is Ramanujan's two-variable theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Nov 04 2005

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
f(a,b) = Sum_{k} a^((k^2+k)/2)*b^((k^2-k)/2) is Ramanujan's two-variable theta function.
|a(n)| is the characteristic function of A093722.
The exponents in the q-series for this sequence are the squares of the numbers of A057538.
This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = -x^4, b = -x.

			1 + x - x^3 - x^7 - x^8 - x^14 + x^20 + x^29 + x^31 + x^42 - x^52 - x^66 - ...
q + q^121 - q^361 - q^841 - q^961 - q^1681 + q^2401 + q^3481 + q^3721 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Quintuple Product Identity

Cf. A010815, A057538, A093722, A113430.

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^5] QPochhammer[ q^2, q^5] QPochhammer[ q^3, q^5])^2 / QPochhammer[ q], {q, 0, n}] (* Michael Somos, Jul 17 2012 *)

{a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k)^((k%5==0) - kronecker( 5, k)), 1 + x * O(x^n)), n))}

{a(n) = n*=5; if( issquare( 24*n + 1, &n), kronecker( 12, n))}

Expansion of f(-x^7, -x^8) + x * f(-x^2, -x^13) where f() is Ramanujan's two-variable theta function.
Euler transform of period 5 sequence [ 1, -1, -1, 1, -1, ...].
G.f.: Sum_{k} (-1)^k * x^(5*k * (3*k + 1)/2) * (x^(-3*k) + x^(3*k + 1)).
G.f.: Product_{k>0} (1 - x^(5*k)) * (1 - x^(5*k - 2)) * (1 - x^(5*k - 3)) / ((1 - x^(5*k - 1)) * (1 - x^(5*k - 4))).
A010815(5*n) = a(n).

A208546 Expansion of f(-x^29, x^31) + x * f(-x^19, x^41) - x^3 * f(-x^11, x^49) + x^7 * f(x, -x^59) in powers of x where f() is Ramanujan's two-variable theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Feb 28 2012

sign

			G.f. = 1 + x - x^3 + x^7 + x^8 + x^14 - x^20 - x^29 + x^31 + x^42 - x^52 - x^66 + ...
G.f. = q + q^121 - q^361 + q^841 + q^961 + q^1681 - q^2401 - q^3481 + q^3721 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A057538, A093722.

f[x_, y_]:= QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; CoefficientList[Series[f[-x^29, x^31] + x*f[-x^19, x^41] - x^3*f[-x^11, x^49] + x^7*f[x, -x^59], {x, 0, 50}], x] (* G. C. Greubel, Aug 11 2018 *)

{a(n) = local(m); if( issquare( 120*n + 1, &m), (-1)^(m \ 40 + m \ 12))}

|a(n)| is the characteristic function of A093722.
The exponents in the q-series q * A(q^120) are the squares of the numbers in A057538.
Euler transform of a period 80 sequence.
G.f.: Sum_{k} (-1)^floor(k/4) * x^(3*k * (5*k + 1)/2) * (x^(4*k + 1) + x^(-16*k + 7)).

A207710 Expansion of f(x) * f(-x^10) / f(-x^2, -x^8) in powers of x where f() is Ramanujan's two-variable theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Feb 19 2012

sign

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

This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = x^4, b = -x.

			1 + x + x^3 - x^7 + x^8 - x^14 - x^20 - x^29 - x^31 - x^42 - x^52 + x^66 + ...
q + q^121 + q^361 - q^841 + q^961 - q^1681 - q^2401 - q^3481 - q^3721 + ...

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
Eric Weisstein's World of Mathematics, Quintuple Product Identity

Cf. A057538, A093722, A113430.

f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A207710[n_] := SeriesCoefficient[f[x^5, -x^10]*f[-x^2, x^3]/f[-x, x^4], {x, 0, n}]; Table[A207710[n], {n,0,50}] (* G. C. Greubel, Jun 18 2017 *)

{a(n) = local(m); if( issquare( 120*n + 1, &m), kronecker( -120, m) * (-1)^(m \ 15))}

Expansion of f(x^5, -x^10) * f(-x^2, x^3) / f(-x, x^4) = f(-x^7, x^8) + x * f(x^2, -x^13) in powers of x where f() is Ramanujan's two-variable theta function.
Euler transform of period 20 sequence [ 1, -1, 1, -1, 1, -2, 1, 0, 1, -2, 1, 0, 1, -2, 1, -1, 1, -1, 1, -1, ...].
|a(n)| is the characteristic function of A093722.
The exponents in the q-series q * A(q^120) are the squares of the numbers in A057538.
G.f.: Prod_{k>0} (1 - (-x)^k) / ((1 - x^(10*k - 2)) * (1 - x^(10*k - 8))).
G.f.: Sum_{k} (-1)^[-k/2] * x^(5*k * (3*k + 1)/2) * (x^(-3*k) + x^(3*k + 1)).
a(7*n + 2) = a(7*n + 4) = a(7*n + 5) = 0. a(n) * (-1)^n = A113430(n).

A207735 Expansion of f(-x^2, x^3)^2 / f(x, -x^2) in powers of x where f() is Ramanujan's two-variable theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Feb 19 2012

sign

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

This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = -x^4, b = x.

			1 - x + x^3 + x^7 - x^8 - x^14 + x^20 - x^29 - x^31 + x^42 - x^52 - x^66 + ...
q - q^121 + q^361 + q^841 - q^961 - q^1681 + q^2401 - q^3481 - q^3721 + ...

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
Eric Weisstein's World of Mathematics, Quintuple Product Identity

Cf. A057538, A093722, A113681.

f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A207735[n_] := SeriesCoefficient[f[x^5, -x^10]*f[-x^2, x^3]/f[x, -x^4], {x, 0, n}]; Table[A207735[n], {n,0,50}] (* G. C. Greubel, Jun 18 2017 *)

{a(n) = local(m); if( issquare( 120*n + 1, &m), (-1)^n * kronecker( 12, m), 0)}

Expansion of f(x^7, -x^8) - x * f(-x^2, x^13) = f(x^5, -x^10) * f(-x^2, x^3) / f(x, -x^4) where f() is Ramanujan's two-variable theta function.
Euler transform of period 20 sequence [ -1, 0, 1, 1, 1, 0, 1, -1, -1, -2, -1, -1, 1, 0, 1, 1, 1, 0, -1, -1, ...].
G.f.: Sum_{k} (-1)^[k/2] * x^(5*k * (3*k + 1)/2) * (x^(-3*k) - x^(3*k + 1)).
|a(n)| is the characteristic function of A093722.
The exponents in the q-series q * A(q^120) are the squares of the numbers in A057538.
a(7*n + 2) = a(7*n + 4) = a(7*n + 5) = 0. a(n) * (-1)^n = A113681(n).

A214529 Expansion of f(x^29, -x^31) - x * f(x^19, -x^41) + x^3 * f(x^11, -x^49) - x^7 * f(-x, x^59) in powers of x where f() is Ramanujan's two-variable theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jul 20 2012

sign

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

			1 - x + x^3 - x^7 + x^8 + x^14 - x^20 + x^29 - x^31 + x^42 - x^52 - x^66 + ...
q - q^121 + q^361 - q^841 + q^961 + q^1681 - q^2401 + q^3481 - q^3721 + ...

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. A057538, A093722, A208546.

a[ n_] := Module[ {m}, If[ n >= 0 && OddQ[ DivisorSigma[ 0, 120 n + 1]], m = Sqrt[ 120 n + 1]; (-1)^(Quotient[ m, 40] + Quotient[ m, 3]), 0]]; Table[a[n], {n, 0, 30}]

{a(n) = local(m); if( issquare( 120*n + 1, &m), (-1)^(m \ 40 + m \ 3))}

|a(n)| is the characteristic function of A093722.
The exponents in the q-series q * A(q^120) are the squares of the numbers in A057538.
Euler transform of a period 80 sequence.
G.f.: Sum_{k} (-1)^(floor((k - 1)/2) + floor(k/4)) * x^(3*k * (5*k + 1)/2) * (x^(4*k + 1) + x^(-16*k + 7)).
a(n) = (-1)^n * A208546(n).

cp's OEIS Frontend

A093722 Integers of the form (k^2 - 1) / 120.

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A057539 Birthday set of order 7, i.e., numbers congruent to +- 1 modulo 2, 3, 4, 5, 6 and 7.

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A057540 Birthday set of order 8: i.e., numbers congruent to +- 1 modulo 2, 3, 4, 5, 6, 7 and 8.

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A057541 Birthday set of order 9: i.e., numbers congruent to +- 1 modulo 2, 3, 4, 5, 6, 7, 8 and 9.

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A113430 Expansion of f(-x, -x^2) * f(-x^10, -x^20) / f(-x^2, -x^8) in powers of x where f(, ) is Ramanujan's general theta function.

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A113681 Expansion of f(-x^2, -x^3)^2 / f(-x, -x^2) in powers of x where f() is Ramanujan's two-variable theta function.

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A208546 Expansion of f(-x^29, x^31) + x * f(-x^19, x^41) - x^3 * f(-x^11, x^49) + x^7 * f(x, -x^59) in powers of x where f() is Ramanujan's two-variable theta function.