A163746 -id:A163746 - cp's OEIS frontend

A122865 Expansion of chi(x) * phi(x^3) * psi(-x^3) in powers of x where chi(), phi(), psi() are Ramanujan theta functions.

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

Offset: 0

Michael Somos, Sep 15 2006

nonn

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

			G.f. = 1 + x + 2*x^3 + 2*x^4 + x^5 + 3*x^8 + 2*x^11 + 2*x^12 + 2*x^13 + ...
G.f. = q + q^4 + 2*q^10 + 2*q^13 + q^16 + 3*q^25 + 2*q^34 + 2*q^37 + ...

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. A002175, A002654, A035154, A113446, A122856, A122864, A163746, A258277.

phi[q_] := EllipticTheta[3, 0, q]; chi[q_] := ((1 - InverseEllipticNomeQ[q]) * InverseEllipticNomeQ[q]/(16*q))^(-1/24); psi[q_] := (1/2)*q^(-1/8)*EllipticTheta[ 2, 0, q^(1/2)]; s = Series[ chi[q]*phi[q^3]*psi[-q^3], {q, 0, 104}]; a[n_] := Coefficient[s, q, n];
(* or *) a[n_] := If[n == 0, 1, Sum[Boole[Mod[d, 4] == 1] - Boole[Mod[d, 4] == 3], {d, Divisors[3n+1]}]]; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Feb 17 2015, after PARI code *)
a[ n_] := If[ n < 0, 0, DivisorSum[ 3 n + 1, KroneckerSymbol[ -4, #] &]]; (* Michael Somos, Sep 02 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^3] QPochhammer[ x^6], {x, 0, n}]; (* Michael Somos, Sep 02 2015 *)

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

{a(n) = my(A, p, e); if(n <0, 0, n = 3*n+1; A = factor(n); 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)))};

{a(n) = if( n<0, 0, n = 3*n+1; sumdiv(n, d, (d%4==1) - (d%4==3)))}; /* Michael Somos, Apr 19 2007 */

Expansion of chi(x) * f(x^3) * f(-x^6) in powers of x where chi(), f() are Ramanujan theta functions. - Michael Somos, Sep 02 2015
Expansion of q^(-1/3) * eta(q^2)^2 * et(q^6)^4 / (eta(q) * eta(q^3) * eta(q^4) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [1, -1, 2, 0, 1, -4, 1, 0, 2, -1, 1, -2, ...]. - Michael Somos, Apr 19 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 2 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A258228. - Michael Somos, Sep 02 2015
a(n) = A002654(3*n + 1) = A035154(3*n + 1) = A113446(3*n + 1) = A122864(3*n + 1) = A163746(3*n + 1).
a(n) = (-1)^n * A258277(n). a(2*n + 1) = A122856(n). - Michael Somos, Sep 02 2015
a(4*n) = A002175(n). a(4*n + 2) = 0. - Michael Somos, Jan 19 2017

A002175 Excess of number of divisors of 12n+1 of form 4k+1 over those of form 4k+3.

Original entry on oeis.org

1, 2, 3, 2, 1, 2, 2, 4, 2, 2, 1, 0, 4, 2, 3, 2, 2, 4, 0, 2, 2, 0, 4, 2, 3, 0, 2, 6, 2, 2, 1, 2, 0, 2, 2, 2, 2, 4, 2, 0, 4, 4, 4, 0, 1, 2, 0, 4, 2, 0, 2, 2, 5, 2, 0, 2, 2, 4, 4, 2, 0, 2, 4, 2, 2, 0, 4, 0, 0, 2, 3, 2, 4, 2, 0, 4, 0, 6, 2, 4, 1, 0, 4, 2, 2, 2, 2, 0, 0, 2, 0, 2, 8, 2, 2, 0, 2, 4, 0, 4, 2, 2, 3, 2, 2

Offset: 0

N. J. A. Sloane

nonn

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

Number of ways to write n as an ordered sum of 2 generalized pentagonal numbers. - Ilya Gutkovskiy, Aug 14 2017

			G.f. = 1 + 2*x + 3*x^2 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + 4*x^7 + 2*x^8 + 2*x^9 + ...
G.f. = q + 2*q^13 + 3*q^25 + 2*q^37 + q^49 + 2*q^61 + 2*q^73 + 4*q^85 + 2*q^97 + ...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Cerkan, Table of n, a(n) for n = 0..10000
J. W. L. Glaisher, On the square of Euler's series, Proc. London Math. Soc., 21 (1889), 182-194.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002654, A008441, A008442, A035154, A035181, A035184, A112301, A113406.
Cf. A113652, A116604, A121363, A121450, A122856, A122864, A122865, A125061.
Cf. A125079, A129447, A129448, A132004, A134013, A134015, A138741, A138746.
Cf. A138950, A138952, A163746, A258210, A258228, A258256, A258279, A258292.

series(mul( ( (1 + q^n)*(1 - q^(3*n))/(1 + q^(3*n)) )^2, n = 1..100), q, 101):
seq(coeftayl(%, q = 0, n), n = 0..100); # Peter Bala, Jan 05 2025

ed[n_]:=Module[{divs=Divisors[12n+1]},Count[divs,?(Mod[#,4] == 1&)]- Count[divs,?(Mod[#,4]==3&)]]; Array[ed,110,0] (* Harvey P. Dale, Jul 01 2012 *)
a[ n_] := If[ n < 0, 0, With[ {m = 12 n + 1}, Sum[ KroneckerSymbol[ 4, d], {d, Divisors[m]}]]]; (* Michael Somos, Apr 23 2014 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2] QPochhammer[ x^3]^2 / (QPochhammer[ x] QPochhammer[ x^6]))^2, {x, 0, n}]; (* Michael Somos, Apr 23 2014 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x^3] / QPochhammer[ x, x^2])^2, {x, 0, n}]; (* Michael Somos, May 25 2015 *)

{a(n) = if( n<0, 0, n = 12*n + 1; sumdiv( n, d, (d%4==1) - (d%4==3)))}; /* Michael Somos, Sep 19 2005 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A)^2 / (eta(x + A) * eta(x^6 + A)))^2, n))}; /* Michael Somos, Jun 02 2012 */

Expansion of (phi(-x^3) / chi(-x))^2 in powers of x where phi(), chi() are Ramanujan theta functions.
Expansion of q^(-1/12) * (eta(q^2) * eta(q^3)^2 / (eta(q) * eta(q^6)))^2 in powers of q. - Michael Somos, Sep 19 2005
Euler transform of period 6 sequence [ 2, 0, -2, 0, 2, -2, ...]. - Michael Somos, Sep 19 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 2 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A258279. - Michael Somos, May 25 2015
From Michael Somos, Jun 02 2012: (Start)
a(n) = A008441(3*n) = A121363(3*n) = A122865(4*n) = A122856(8*n).
a(n) = A116604(6*n) = A125079(6*n) = A129447(6*n) = A138741(6*n).
a(n) = A(12*n+1) where A = A002654, A008442, A035154, A035181, A035184, A112301, A113406, A113652, A121450, A122864, A125061, A129448, A132004, A134013, A134015, A138746, A138950, A138952, A163746.  (End)
From Michael Somos, May 25 2015: (Start)
a(n) = A258277(4*n) = A258278(8*n) = A258291(3*n).
a(n) = - A258210(12*n + 1) = A258228(12*n + 1) = A258256(12*n + 1).
2*a(n) = A258279(12*n + 1) = - A258292(12*n + 1). (End)
G.f.: (Sum_{k = -oo..oo} x^(k*(3*k-1)/2))^2. - Ilya Gutkovskiy, Aug 14 2017
G.f.: ( Product_{n >= 1} (1 + q^n)*(1 - q^(3*n))/(1 + q^(3*n)) )^2. - Peter Bala, Jan 05 2025

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

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

A122865 Expansion of chi(x) * phi(x^3) * psi(-x^3) in powers of x where chi(), phi(), psi() are Ramanujan theta functions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A002175 Excess of number of divisors of 12n+1 of form 4k+1 over those of form 4k+3.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

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

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