A122857 -id:A122857 - cp's OEIS frontend

A035154 a(n) = Sum_{d|n} Kronecker(-36, d).

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

Offset: 1

N. J. A. Sloane

			G.f. = x + x^2 + x^3 + x^4 + 2*x^5 + x^6 + x^8 + x^9 + 2*x^10 + x^12 + 2*x^13 + ...

Bruce C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, 1994, see p. 197, Entry 44.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A002654, A008441, A019670, A122856, A122857, A122865, A125079.

a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -36, d], { d, Divisors[ n]}]]; (* Michael Somos, Jun 24 2011 *)
a[ n_] := SeriesCoefficient[ (-2 + EllipticTheta[ 3, 0, q]^2 + EllipticTheta[ 3, 0, q^3]^2) / 4, {q, 0, n}]; (* Michael Somos, Jul 09 2013 *)

{a(n) = if( n<1, 0, sumdiv(n, d, kronecker( -36, d)))}; /* Michael Somos, Jul 30 2006 */

{a(n) = if( n<1, 0, direuler(p=2, n, 1 / ((1 - X) * (1 - kronecker( -36, p) * X))) [n])}; /* Michael Somos, Jul 30 2006 */

{a(n)=polcoeff(sum(m=0,n\6+1,(-1)^m*(x^(6*m+1)/(1-x^(6*m+1)+x*O(x^n)) + x^(6*m+5)/(1-x^(6*m+5)+x*O(x^n)))),n)} /* Paul D. Hanna */

Expansion of -1 + (theta_3(q)^2 + theta_3(q^3)^2) / 2 in powers of q. - Michael Somos, Jul 09 2013
From Michael Somos, Jul 30 2006: (Start)
Moebius transform is period 12 sequence [1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, ...].
Multiplicative with a(2^e) = a(3^e) = 1, a(p^e) = e+1 if p == 1(mod 4), a(p^e) = (1 + (-1)^e) / 2 if p == 3(mod 4). (End)
Dirichlet g.f.: zeta(s) * L(chi,s) where chi(n) = Kronecker( -36, n). Sum_{n>0} a(n) / n^s = Product_{p prime} 1 / ((1 - p^-s) * (1 - Kronecker( -36, p) * p^-s)). - Michael Somos, Jun 24 2011
a(2*n) = a(3*n) = a(n). a(2*n + 1) = A125079(n). a(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n). a(4*n + 1) = A008441(n).
2 * a(n) = A122857(n) unless n=0. - Michael Somos, Jul 09 2013
G.f.: Sum_{n>=0} (-1)^n*( x^(6*n+1)/(1-x^(6*n+1)) + x^(6*n+5)/(1-x^(6*n+5)) ). - Paul D. Hanna, Dec 14 2011
G.f.: x/(1-x) + x^5/(1-x^5) - x^7/(1-x^7) - x^11/(1-x^11) + x^13/(1-x^13) + x^17/(1-x^17) --++ ...
a(n) = A002654(n) + A002654(3*n). - Michael Somos, Jan 25 2017
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 = 1.0471975... (A019670). - Amiram Eldar, Nov 17 2023

A125061 Expansion of psi(q) * psi(q^2) * chi(q^3) * chi(-q^6) in powers of q where psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 3, 0, 1, 1, 2, 0, 3, 2, 0, 6, 1, 2, 1, 0, 2, 0, 0, 0, 3, 3, 2, 3, 0, 2, 6, 0, 1, 0, 2, 0, 1, 2, 0, 6, 2, 2, 0, 0, 0, 2, 0, 0, 3, 1, 3, 6, 2, 2, 3, 0, 0, 0, 2, 0, 6, 2, 0, 0, 1, 4, 0, 0, 2, 0, 0, 0, 1, 2, 2, 9, 0, 0, 6, 0, 2, 1, 2, 0, 0, 4, 0, 6, 0, 2, 2, 0, 0, 0, 0, 0, 3, 2, 1, 0, 3, 2, 6, 0, 2

Offset: 0

Michael Somos, Nov 18 2006

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

			G.f. = 1 + q + q^2 + 3*q^3 + q^4 + 2*q^5 + 3*q^6 + q^8 + q^9 + 2*q^10 + 3*q^12 + ...

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 82, Eq. (32.53).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A002175, A008441, A019669, A053692, A113407, A121444, A122856, A122857, A122865, A125061, A138741, A138745, A138746, A138949, A138950, A138951, A138952, A163746.

Cf. A000122, A000700, A010054, A121373.

s = (EllipticTheta[3, 0, q]^2 + 3*EllipticTheta[3, 0, q^3]^2)/4 + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Dec 07 2015, from 2nd formula *)

{a(n) = if( n<1, n==0, sumdiv(n, d, ((d%2) * ((d%3==0)+1)) * (-1)^(d\6)))};

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); prod( k=1, matsize(A)[1],
     [p, e] = A[k, ]; if( p==2, 1, p==3, 1+e%2*2, p%4==1, e+1, !(e%2) )))};

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

Expansion of eta(q^2) * eta(q^4)^2 * eta(q^6)^3 / (eta(q) * eta(q^3) * eta(q^12)^2) in powers of q.
Expansion of (theta_3(q)^2 + 3*theta_3(q^3)^2) / 4 in powers of q.
Euler transform of period 12 sequence [ 1, 0, 2, -2, 1, -2, 1, -2, 2, 0, 1, -2, ...].
Moebius transform is period 12 sequence [ 1, 0, 2, 0, 1, 0, -1, 0, -2, 0, -1, 0, ...].
a(n) is multiplicative with a(2^e) = 1, a(3^e) = 2-(-1)^e, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) == (1-(-1)^e)/2 if p == 3 (mod 4).
G.f.: 1 + Sum_{k>0} (x^k + x^(3*k)) / (1 - x^(2*k) + x^(4*k)).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 3 (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A122857.
a(12*n + 7) = a(12*n + 11) = 0. a(2*n) = a(n). a(2*n + 1) = A138741(n). a(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n). a(4*n + 1) = A008441(n). a(4*n + 3) = 3 * A008441(n). a(6*n + 1) = A002175(n). a(6*n + 5) = 2 * A121444(n). a(8*n + 1) = A113407(n). a(8*n + 3) = 3 * A113407(n). a(8*n + 5) = 2 * A053692(n). a(8*n + 7) = 6 * A053692(n). a(9*n) = A125061(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 (A019669). - Amiram Eldar, Nov 24 2023

A281451 Expansion of x * f(x, x) * f(x, x^17) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Jan 23 2017

nonn

			G.f. = x + 3*x^2 + 2*x^3 + 2*x^5 + 2*x^6 + 2*x^10 + 2*x^11 + 2*x^17 + ...
G.f. = q^16 + 3*q^25 + 2*q^34 + 2*q^52 + 2*q^61 + 2*q^97 + 2*q^106 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A002654, A004018, A019670, A035154, A105673, A113446, A122856, A122857, A122864, A122865.
Cf. A125061, A129448, A138949, A138950, A163746, A256269, A256276, A256280, A258034.
Cf. A258228, A258256, A258278, A258292, A259761.
Cf. A281452, A281453, A281490, A281491, A281492.

a[ n_] := If[ n < 0, 0, DivisorSum[ 9 n + 7, KroneckerSymbol[ -4, #] &]];
a[ n_] := If[ n < 0, 0, Times @@ (Which[# < 3, 1, Mod[#, 4] == 1, #2 + 1, True, (1 + (-1)^#2) / 2] & @@@ FactorInteger[ 9 n + 7])];
a[ n_] := SeriesCoefficient[ x EllipticTheta[ 3, 0, x] QPochhammer[ -x, x^18] QPochhammer[ -x^17, x^18] QPochhammer[ x^18], {x, 0, n}];

{a(n) = if( n<0, 0, sumdiv(9*n + 7, d, (d%4==1) - (d%4==3)))};

{a(n) = if( n<0, 0, my(A, p, e); A = factor(9*n + 7); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, 1, p%4==1, e+1, 1-e%2)))};

{a(n) = if( n<0, 0, my(m = 9*n + 7, k, s); forstep(j=0, sqrtint(m), 3, if( issquare(m - j^2, &k) && (k%9 == 4 || k%9 == 5), s+=(j>0)+1)); s)};

f(x,x^m) = 1 + Sum_{k>=1} x^((m+1)*k*(k-1)/2) (x^k + x^(m*k)). - N. J. A. Sloane, Jan 30 2017
Euler transform of a period 36 sequence.
G.f.: x * (Sum_{k in Z} x^k^2) * (Sum_{k in Z} x^(9*k^2 + 8*k)).
G.f.: x * Product_{k>0} (1 + x^(2*k-1))^2 * (1 - x^(2*k)) * (1 + x^(18*k-17)) * (1 + x^(18*k-1)) * (1 - x^(18*k)).
a(4*n) = a(8*n + 7) = a(16*n + 13) = a(32*n + 9) = a(49*n + 7) = a(98*n + 14) = 0.
a(4*n + 1) = A281452(n).  a(8*n + 3) = 2 * A281491(n).  A(16*n + 1) = A281453(n).
a(32*n + 25) = 2 * A281490(n). a(64*n + 49) = a(n). a(128*n + 17) = 2 * A281492(n).
a(n) = A122865(3*n + 2). a(n) = A122856(6*n + 4) = A258278(6*n + 4).
a(n) = b(9*n + 7) where b = A002654, A035154, A113446, A122864, A125061, A129448, A138950, A163746, A256276, A258228, A258256.
2 * a(n) = b(9*n + 7) where b = A105673, A122857, A258034, A259761.  -2 * a(n) = b(9*n + 7) where b = A138949, A256280, A258292.
a(n) = - A256269(9*n + 7). 4 * a(n) = A004018(9*n + 7).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 = 1.0471975... (A019670). - Amiram Eldar, Jan 20 2025

A132003 Expansion of (phi(q^3) / phi(q)) * phi(-q^2) * phi(-q^6) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 2, -2, 2, -4, 2, 0, 2, -2, 4, 0, 2, -4, 0, -4, 2, -4, 2, 0, 4, 0, 0, 0, 2, -6, 4, -2, 0, -4, 4, 0, 2, 0, 4, 0, 2, -4, 0, -4, 4, -4, 0, 0, 0, -4, 0, 0, 2, -2, 6, -4, 4, -4, 2, 0, 0, 0, 4, 0, 4, -4, 0, 0, 2, -8, 0, 0, 4, 0, 0, 0, 2, -4, 4, -6, 0, 0, 4, 0

Offset: 0

Michael Somos, Aug 06 2007

sign

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

			G.f. = 1 - 2*x + 2*x^2 - 2*x^3 + 2*x^4 - 4*x^5 + 2*x^6 + 2*x^8 - 2*x^9 + 4*x^10 + ...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Equation (32.72).

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A008441, A122857, A125079, A132004.

a[ n_] := If[ n < 1, Boole[n == 0], -2 DivisorSum[ n, (-1)^(n + #) KroneckerSymbol[ -36, #] &]]; (* Michael Somos, Nov 01 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], -2 Times @@ (Which[ # < 5, -(-1)^#, Mod[#, 4] == 3, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger @ n)]; (* Michael Somos, Nov 01 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^3] EllipticTheta[ 4, 0, q^2] EllipticTheta[ 4, 0, q^6] / EllipticTheta[ 3, 0, q], {q, 0, n}]; (* Michael Somos, Nov 01 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q]^2 + EllipticTheta[ 4, 0, q^3]^2) / 2, {q, 0, n}]; (* Michael Somos, Mar 05 2023 *)

{a(n) = if( n<1, n==0, -2 * sumdiv(n, d, (-1)^(n+d) * kronecker(-36, d)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)^7 / (eta(x^2 + A)^3 * eta(x^3 + A)^2 * eta(x^12 + A)^3), n))};

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); -2 * prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 1, p==2, -1, p%4==1, e+1, 1-e%2)))};

Expansion of eta(q)^2 * eta(q^4) * eta(q^6)^7 / (eta(q^2)^3 * eta(q^3)^2 * eta(q^12)^3) in powers of q.
a(n) = -2*b(n) where b() is multiplicative with b(2^e) = 2*0^e - 1, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 4), b(p^e) = (1 + (-1)^e)/2 if p == 3 (mod 4).
Euler transform of period 12 sequence [-2, 1, 0, 0, -2, -4, -2, 0, 0, 1, -2, -2, ...].
G.f.: 1 - 2 * Sum_{k>0} Kronecker(-36, k) * x^k / (1 + x^k).
a(n) = - A132004(n) unless n=0.
a(2*n) = A122857(n). a(2*n + 1) = -2 * A125079(n). a(3*n) = a(n). a(3*n + 1) = -2 * A258277(n). a(3*n + 2) = 2 * A258278(n). - Michael Somos, Nov 01 2015
a(12*n + 7) = a(12*n + 11) = 0. a(4*n + 1) = -2 * A008441(n).
a(n) = (-1)^n * A122857(n). Expansion of (phi(-q)^2 + phi(-q^3)^2) / 2 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Mar 05 2023

A281452 Expansion of f(x, x) * f(x^5, x^13) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jan 26 2017

nonn

			G.f. = 1 + 2*x + 2*x^4 + x^5 + 2*x^6 + 4*x^9 + x^13 + 4*x^14 + 2*x^16 + ...
G.f. = q^4 + 2*q^13 + 2*q^40 + q^49 + 2*q^58 + 4*q^85 + q^121 + 4*q^130 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A002654, A004018, A019670, A035154, A105673, A113446, A122856, A122857, A122864, A122865.
Cf. A125061, A129448, A138949, A138950, A163746, A256269, A256276, A256280, A258034.
Cf. A258228, A258256, A258278, A258292, A259761.
Cf. A281451, A281453, A281490, A281491, A281492.

a[ n_] := If[ n < 0, 0, DivisorSum[ 9 n + 4, KroneckerSymbol[ -4, #] &]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^5, x^18] QPochhammer[ -x^13, x^18] QPochhammer[ x^18], {x, 0, n}];
a[ n_] := If[ n < 0, 0, Times @@ (Which[ # < 3, 1, Mod[#, 4] == 1, #2 + 1, True, (1 + (-1)^#2) / 2] & @@@ FactorInteger[ 9 n + 4])];

{a(n) = if( n<0, 0, sumdiv(9*n + 4, d, (d%4==1) - (d%4==3)))};

{a(n) = if( n<0, 0, my(m = 9*n + 4, k, s); forstep(j=0, sqrtint(m), 3, if( issquare(m - j^2, &k) && (k%9 == 2 || k%9 == 7), s+=(j>0)+1)); s)};

{a(n) = if( n<0, 0, my(A, p, e); A = factor(9*n + 4); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, 1, p%4==1, e+1, 1-e%2)))};

f(a,b) = 1 + Sum_{k=1..oo} (ab)^(k(k-1)/2)*(a^k+b^k). - N. J. A. Sloane, Jan 30 2017
Euler transform of a period 36 sequence.
G.f.: (Sum_{k in Z} x^k^2) * (Sum_{k in Z} x^(9*k^2 + 4*k)).
G.f.: Product_{k>0} (1 + x^(2*k-1))^2 * (1 - x^(2*k)) * (1 - x^(18*k-13)) * (1 - x^(18*k-5)) * (1 - x^(18*k)).
a(n) = A122865(3*n + 1) = A122856(6*n + 2) = A258278(6*n + 2). a(n) = - A256269(9^n + 4). 4 * a(n) = A004018(9*n + 4).
a(n) = b(9*n + 4) with b = A002654, A035154, A113446, A122864, A125061, A129448, A138950, A163746, A256276, A258228, A258256.
2 * a(n) = b(9*n + 4) = with b = A105673, A105673, A122857, A258034, A259761. -2 * a(n) = b(9*n + 4) with b = A138949, A256280, A258292.
a(4*n) = A281453(n). a(8*n + 6) = 2 * A281490(n). a(16*n + 12) = A281451(n).
a(32*n + 4) = 2 * A281492(n). a(64*n + 28) = A281452(n). a(128*n + 60) = 2 * A281491(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 = 1.0471975... (A019670). - Amiram Eldar, Jan 20 2025

A281453 Expansion of f(x, x) * f(x^7, x^11) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 2, 0, 0, 2, 0, 0, 1, 2, 2, 0, 3, 2, 0, 0, 2, 4, 0, 0, 0, 2, 0, 0, 2, 0, 2, 0, 2, 0, 0, 0, 0, 3, 2, 0, 0, 6, 0, 0, 0, 1, 4, 0, 2, 2, 0, 0, 2, 2, 4, 0, 0, 0, 0, 0, 0, 4, 2, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 2, 0, 0, 0, 3, 4, 0, 0, 2, 0, 4, 0, 0, 2, 0, 0

Offset: 0

Michael Somos, Jan 26 2017

nonn

			G.f. = 1 + 2*x + 2*x^4 + x^7 + 2*x^8 + 2*x^9 + 3*x^11 + 2*x^12 + ...
G.f. = q + 2*q^10 + 2*q^37 + q^64 + 2*q^73 + 2*q^82 + 3*q^100 + ...

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. A000122, A002654, A004018, A035154, A091400, A105673, A113446, A122856, A122857.
Cf. A122864, A122865, A125061, A129448, A138949, A138950, A163746, A205808, A256276.
Cf. A256280, A258034, A258228, A258256, A258278, A258292, A259761.
Cf. A281451, A281452, A281490, A281491, A281492.

a[ n_] := If[ n < 0, 0, DivisorSum[ 9 n + 1, KroneckerSymbol[ -4, #] &]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^7, x^18] QPochhammer[ -x^11, x^18] QPochhammer[ x^18], {x, 0, n}];
a[ n_] := If[ n < 0, 0, Times @@ (Which[# < 3, 1, # == 3, Mod[#2, 2] 2 + 1, Mod[#, 4] == 1, #2 + 1, True, (1 + (-1)^#2) / 2] & @@@ FactorInteger[ 9 n + 1])];

{a(n) = if( n<0, 0, sumdiv(9*n + 1, d, kronecker(-4, d)))};

{a(n) = if( n<0, 0, my(m = 9*n + 1, k, s); forstep(j=0, sqrtint(m), 3, if( issquare(m - j^2, &k) && (k%9 == 1 || k%9 == 8), s+=(j>0)+1)); s)};

{a(n) = if( n<0, 0, my(A, p, e); A = factor(9*n + 1); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, 1, p==3, -2*(-1)^e, p%4==1, e+1, 1-e%2)))};

f(a,b) = 1 + Sum_{k=1..oo} (ab)^(k(k-1)/2)*(a^k+b^k). - N. J. A. Sloane, Jan 30 2017
Euler transform of a period 36 sequence.
G.f.: (Sum_{k in Z} x^k^2) * (Sum_{k in Z} x^(9*k^2 + 2*k)).
G.f.: Product_{k>0} (1 + x^(2*k-1))^2 * (1 - x^(2*k)) * (1 + x^(18*k-11)) * (1 + x^(18*k-7)) * (1 - x^(18*k)).
a(4*n + 2) = a(8*n + 5) = a(16*n + 3) = a(32*n + 31) = a(64*n + 55) = a(128*n + 39) = 0.
a(4*n + 3) = A281451(n). a(8*n + 1) = 2 * A281492(n). a(16*n + 7) = A281452(n). a(32*n + 15) = 2 * A281491(n). a(128*n + 103) = 2 * A281490(n).
a(n) = A122865(3*n) = A122856(6*n) = A258278(6*n) = a(64*n + 7). a(n) = -A256269(9*n + 1).
a(n) = b(9*n + 1) where b = A002654, A035154, A091400, A113446, A122864, A125061, A129448, A138950, A163746, A256276, A258228, A258256.
2 * a(n) = b(9*n + 1) where b = A105673, A122857, A258034, A259761. 2 * a(n) = - b(9*n+1) where b = A138949, A256280, A258292. 4 * a(n) = A004018(9*n + 1).
Convolution of A000122 and A205808.

cp's OEIS Frontend

A035154 a(n) = Sum_{d|n} Kronecker(-36, d).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A125061 Expansion of psi(q) * psi(q^2) * chi(q^3) * chi(-q^6) in powers of q where psi(), chi() are Ramanujan theta functions.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A281451 Expansion of x * f(x, x) * f(x, x^17) in powers of x where f(, ) is Ramanujan's general theta function.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A132003 Expansion of (phi(q^3) / phi(q)) * phi(-q^2) * phi(-q^6) in powers of q where phi() is a Ramanujan theta function.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A281452 Expansion of f(x, x) * f(x^5, x^13) in powers of x where f(, ) is Ramanujan's general theta function.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A281453 Expansion of f(x, x) * f(x^7, x^11) in powers of x where f(, ) is Ramanujan's general theta function.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica