A192013 -id:A192013 - cp's OEIS frontend

A279541 Indices of records in A192013: Sum_{d|n} Kronecker(-6, d).

1, 5, 25, 35, 175, 385, 1225, 1925, 9625, 13475, 48125, 55825, 148225, 279125, 390775, 1395625, 1730575, 4298525, 8652875, 12114025, 43264375, 60570125, 133254275, 302850625, 423990875, 642043325, 2119954375

Offset: 1

Charles R Greathouse IV, Dec 14 2016

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..400

Cf. A192013, A279542, A109017.

smooth(P:vec, lim)=my(v=List([1]), nxt=vector(#P, i, 1), indx, t); while(1, t=vecmin(vector(#P, i, v[nxt[i]]*P[i]), &indx); if(t>lim, break); if(t>v[#v], listput(v, t)); nxt[indx]++); Vec(v);
ok(p)=p=p%24; p==1 || p==5 || p==7 || p==11;
A192013(n)=sumdivmult(n, d, kronecker(-6, d));
list(lim)=my(u=List([1]),v=[5],r,t); forprime(p=7,, if(ok(p), if(v[#v]*p>lim, break); v=concat(v, v[#v]*p))); v=smooth(v,lim); for(i=2,#v, t=A192013(v[i]); if(t>r, r=t; listput(u,v[i]))); Vec(u)

A279542 Records in A192013: Sum_{d|n} Kronecker(-6, d).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 20, 24, 27, 32, 36, 40, 48, 54, 64, 72, 80, 96, 108, 120, 128, 144, 160, 192, 216, 240, 256, 288, 320, 360, 384, 432, 480, 512, 576, 640, 720, 768, 864, 960, 1024, 1152, 1280, 1440, 1536, 1728, 1920, 2048, 2304, 2400, 2560, 2880, 3072, 3456, 3840, 4096

Offset: 1

Charles R Greathouse IV, Dec 14 2016

nonn

			Let (m|n) = Kronecker(m, n) be the Kronecker symbol.
A192013(1) = (-6|1) = 1 = a(1).
A192013(5) = (-6|1) + (-6|5) = 2 = a(2).
A192013(25) = (-6|1) + (-6|5) + (-6|25) = 3 = a(3).
A192013(35) = (-6|1) + (-6|5) + (-6|7) + (-35|1) = 4 = a(4).
A192013(175) = (-6|1) + (-6|5) + (-6|7) + (-6|25) + (-6|35) + (-6|175) = 6 = a(5).
A192013(385) = (-6|1) + (-6|5) + (-6|7) + (-6|11) + (-6|35) + (-6|55) + (-6|77) + (-6|385) = 8 = a(6).

Charles R Greathouse IV, Table of n, a(n) for n = 1..400

Cf. A192013, A279541, A109017, A000005.

a(n) = A192013(A279541(n)) = A000005(A279541(n)).

A115660 Expansion of (phi(q) * phi(q^6) - phi(q^2) * phi(q^3)) / 2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Jan 28 2006

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

Number 41 of the 74 eta-quotients listed in Table I of Martin (1996). - Michael Somos, Mar 14 2012

			G.f. = q - q^2 - q^3 + q^4 - 2*q^5 + q^6 + 2*q^7 - q^8 + q^9 + 2*q^10 - 2*q^11 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
A. Berkovich and H. Yesilyurt, Ramanujan's identities and representation of integers by certain binary and quaternary quadratic forms, arXiv:math/0611300 [math.NT], 2006-2007.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000377, A046113, A108563, A128580, A128581, A192013, A259668, A260308, A261115, A263548, A263571.

a[ n_] := SeriesCoefficient[ q QPochhammer[ q] QPochhammer[ q^4] QPochhammer[ q^6] QPochhammer[ q^24] / (QPochhammer[ q^3] QPochhammer[ q^8]), {q, 0, n}]; (* Michael Somos, Apr 19 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^6] - EllipticTheta[ 3, 0, q^2] EllipticTheta[ 3, 0, q^3]) / 2, {q, 0, n}]; (* Michael Somos, Apr 19 2015 *)
a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ 2, d] KroneckerSymbol[ -3, n/d], {d, Divisors[ n]}]]; (* Michael Somos, Apr 19 2015 *)
a[ n_] := If[ n < 1, 0, Times @@ (Which[ # == 1, 1, # < 5, (-1)^#2, Mod[#, 24] < 12, (#2 + 1) KroneckerSymbol[ #, 12]^#2, True, 1 - Mod[#2, 2]]& @@@ FactorInteger[n])]; (* Michael Somos, Oct 22 2015 *)

{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<5, (-1)^e, p%24<12, (e+1) * kronecker( p, 12)^e, 1-e%2)))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^24 + A) / (eta(x^3 + A) * eta(x^8 + A)), n))};

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

Expansion of eta(q) * eta(q^4) * eta(q^6) * eta(q^24) / (eta(q^3) * eta(q^8)) in powers of q.
Euler transform of period 24 sequence [ -1, -1, 0, -2, -1, -1, -1, -1, 0, -1, -1, -2, -1, -1, 0, -1, -1, -1, -1, -2, 0, -1, -1, -2, ...].
a(n) is multiplicative with a(2^e) = a(3^e) = (-1)^e, a(p^e) = e+1 if p == 1, 7 (mod 24), a(p^e) = (e+1) * (-1)^e if p == 5, 11 (mod 24), a(p^e) = (1 + (-1)^e) / 2 if p == 13, 17, 19, 23 (mod 24).
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 24^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
G.f.: Sum_{k>0} Kronecker(k,8) * x^k / (1 + x^k + x^(2*k)) = Sum_{k>0} Kronecker(k,3) * x^k * (1 - x^(2*k)) / (1 + x^(4*k)).
abs(a(n)) = A000377(n). a(n) = (-1)^n * A128581(n). a(2*n) = a(3*n) = -a(n). a(2*n + 1) = A128580(n). - Michael Somos, Mar 14 2012
abs(a(n)) = A192013(n) unless n=0. - Michael Somos, Oct 22 2015
a(3*n + 1) = A263571(n). a(4*n) = A259668(n). a(6*n + 1) = A261115(n). a(6*n + 4) = A263548(n). a(8*n + 1) = A260308(n). - Michael Somos, Oct 22 2015
a(n) = A000377(n) - A108563(n) = A046113(n) - A000377(n). - Michael Somos, Oct 22 2015

A128581 Expansion of (phi(q^2) * phi(-q^3) - phi(-q) * phi(q^6)) / 2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Mar 11 2007

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

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

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. A115660, A128580.

a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[ n, KroneckerSymbol[ #, 8] KroneckerSymbol[ n/#, 3] &]]; (* Michael Somos, Nov 15 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q^2] EllipticTheta[ 4, 0, q^3] - EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^6])/2, {q, 0, n}]; (* Michael Somos, Nov 15 2015 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ -q] QPochhammer[ q^24] QPochhammer[ q^4, q^8] QPochhammer[ q^3, -q^3], {q, 0, n}]; (* Michael Somos, Nov 15 2015 *)

{a(n) = -(-1)^n * sumdiv(n, d, kronecker(d, 8) * kronecker(n/d, 3))}

{a(n) = my(A =  x * O(x^n)); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^12 + A) * eta(x^24 + A) / (eta(x + A) * eta(x^6 + A)^2 * eta(x^8 + A)), n)}

A128581(n)={prod(i=1,matsize(n=factor(n))[1], if(12>n[i,1]%24, if(bittest(12,n[i,1]),(-1)^(n[i,1]+n[i,2]-1), if(bittest(n[i,2],0)&&n[i,1]%6>1,-n[i,2]-1,n[i,2]+1)), !bittest(n[i,2],0)))} \\ a(p^e) as given in formula. - M. F. Hasler, May 07 2018

Expansion of eta(q^2)^3 * eta(q^3) * eta(q^12) * eta(q^24) / (eta(q) * eta(q^6)^2 * eta(q^8)) in powers of q.
Euler transform of period 24 sequence [ 1, -2, 0, -2, 1, -1, 1, -1, 0, -2, 1, -2, 1, -2, 0, -1, 1, -1, 1, -2, 0, -2, 1, -2, ...].
Multiplicative with a(2^e) = -(-1)^e if e>0, a(3^e) = (-1)^e, a(p^e) = e+1 if p == 1, 7 (mod 24), a(p^e) = (e+1)(-1)^e if p == 5, 11 (mod 24), a(p^e) = (1 + (-1)^e)/2 if p == 13, 17, 19, 23 (mod 24).
a(n) = -(-1)^n * A115660(n). a(2*n) = A115660(n). a(2*n + 1) = A128580(n).
abs(a(n)) = A000377(n) = A192013(n). - M. F. Hasler, May 07 2018

A263548 Expansion of f(x, x) * f(x^2, x^10) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Oct 20 2015

nonn

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

			G.f. = 1 + 2*x + x^2 + 2*x^3 + 2*x^4 + 2*x^6 + 2*x^9 + x^10 + 4*x^11 + ...
G.f. = q^2 + 2*q^5 + q^8 + 2*q^11 + 2*q^14 + 2*q^20 + 2*q^29 + q^32 + ...

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. A000377, A115660, A192013.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^2, x^12] QPochhammer[ -x^10, x^12] QPochhammer[ x^12], {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^2, x^4] EllipticTheta[ 2, Pi/4, x^3] / (2^(1/2) x^(3/4)), {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^6 + A) * eta(x^24 + A) / (eta(x + A)^2 * eta(x^8 + A) * eta(x^12 + A)), n))};

Expansion of phi(x) * chi(x^2) * psi(-x^6) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of q^(-2/3) * eta(q^2)^4 * eta(q^6) * eta(q^24) / (eta(q)^2 * eta(q^8) * eta(q^12)) in powers of q.
Euler transform of period 24 sequence [ 2, -2, 2, -2, 2, -3, 2, -1, 2, -2, 2, -2, 2, -2, 2, -1, 2, -3, 2, -2, 2, -2, 2, -2, ...].
a(n) = A000377(3*n + 2) = A192013(3*n + 2) = A115660(6*n + 4).

A035184 a(n) = Sum_{d|n} Kronecker(-1, d).

Original entry on oeis.org

1, 2, 0, 3, 2, 0, 0, 4, 1, 4, 0, 0, 2, 0, 0, 5, 2, 2, 0, 6, 0, 0, 0, 0, 3, 4, 0, 0, 2, 0, 0, 6, 0, 4, 0, 3, 2, 0, 0, 8, 2, 0, 0, 0, 2, 0, 0, 0, 1, 6, 0, 6, 2, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 7, 4, 0, 0, 6, 0, 0, 0, 4, 2, 4, 0, 0, 0, 0, 0, 10, 1, 4, 0, 0, 4, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 2, 2, 0, 9, 2, 0, 0, 8, 0

Offset: 1

N. J. A. Sloane

			G.f. = x + 2*x^2 + 3*x^4 + 2*x^5 + 4*x^8 + x^9 + 4*x^10 + 2*x^13 + 5*x^16 + 2*x^17 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A002175, A008441, A019669, A053692, A113407, A121444.
Inverse Moebius transform of A034947.
Sum_{d|n} Kronecker(k, d): A035143..A035181 (k=-47..-9, skipping numbers that are not cubefree), A035182 (k=-7), A192013 (k=-6), A035183 (k=-5), A002654 (k=-4), A002324 (k=-3), A002325 (k=-2), this sequence (k=-1), A000012 (k=0), A000005 (k=1), A035185 (k=2), A035186 (k=3), A001227 (k=4), A035187..A035229 (k=5..47, skipping numbers that are not cubefree).

a[n_] := DivisorSum[n, KroneckerSymbol[-1, #] &]; Array[a, 105] (* Jean-François Alcover, Dec 02 2015 *)

{a(n) = if( n<1, 0, direuler( p=2, n, 1/((1 - X) * (1 - kronecker( -1, p) * X))) [n])}; /* Michael Somos, Jan 05 2012 */

{a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -1, d)))}; /* Michael Somos, Jan 05 2012 */

a(n) is multiplicative with a(2^e) = e + 1, a(p^e) = e + 1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4). - Michael Somos, Jan 05 2012
a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0. a(4*n + 1) = A008441(n). a(8*n + 1) = A113407(n). a(8*n + 5) = 2 * A053692(n). a(12*n + 1) = A002175(n). a(12*n + 5) = 2 * A121444(n).
Dirichlet g.f.: zeta(s)*beta(s)/(1 - 2^(-s)), where beta is the Dirichlet beta function. - Ralf Stephan, Mar 27 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 = 1.570796... (A019669). - Amiram Eldar, Oct 17 2022

A257920 Expansion of phi(x) * psi(x^3) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jul 12 2015

nonn

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

			G.f. = 1 + 2*x + x^3 + 4*x^4 + 2*x^7 + 3*x^9 + 2*x^10 + 2*x^12 + 2*x^13 + ...
G.f. = q^3 + 2*q^11 + q^27 + 4*q^35 + 2*q^59 + 3*q^75 + 2*q^83 + 2*q^99 + ...

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. A000377, A128591, A129402, A134177, A192013.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 2, 0, x^(3/2)] / (2 x^(3/8)), {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^6 + A)^2 / (eta(x + A)^2 * eta(x^3 + A) * eta(x^4 + A)^2), n))};

Expansion of q^(-3/8) * eta(q^2)^5 * eta(q^6)^2 / (eta(q)^2 * eta(q^3) * eta(q^4)^2) in powers of q.
Euler transform of period 12 sequence [ 2, -3, 3, -1, 2, -4, 2, -1, 3, -3, 2, -2, ...].
a(n) = A129402(4*n + 1) = A134177(4*n + 1) = A000377(8*n + 3) = A192013(8*n + 3).
a(3*n + 2) = 0. a(3*n + 1) = 2 * A128591(n).

A259895 Expansion of psi(x^2) * psi(x^3) in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jul 07 2015

nonn

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

			G.f. = 1 + x^2 + x^3 + x^5 + x^6 + 2*x^9 + x^11 + x^12 + 2*x^15 + x^18 + ...
G.f. = q^5 + q^21 + q^29 + q^45 + q^53 + 2*q^77 + q^93 + q^101 + 2*q^125 + ...

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. A000377, A128580, A128583, A129402, A134177, A190615, A192013, A257921, A259668, A259896.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] EllipticTheta[ 2, 0, x^(3/2)] / (4 q^(5/8)), {x, 0, n}];
a[ n_] := If[ n < 0, 0, 1/2 Sum[ KroneckerSymbol[ -6, d], {d, Divisors[8 n + 5]}]]; (* Michael Somos, Jul 22 2015 *)

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

{a(n) = if( n<0, 0, 1/2 * sumdiv( 8*n + 5, d, kronecker( -6, d)))};

Expansion of q^(-5/8) * eta(q^4)^2 * eta(q^6)^2 / (eta(q^2) * eta(q^3)) in powers of q.
Euler transform of period 12 sequence [ 0, 1, 1, -1, 0, 0, 0, -1, 1, 1, 0, -2, ...].
a(n) = A259896(3*n + 1). a(3*n) = A128583(n). a(3*n + 1) = a(9*n + 8) = 0.
2 * a(n) = A129402(4*n + 2) = A190615(4*n + 2) = A000377(8*n + 5) = A192013(8*n + 5). - Michael Somos, Jul 22 2015
-2 * a(n) = A259668(2*n + 1) = A128580(4*n + 2) = A134177(4*n + 2) = A257921(6*n + 3). - Michael Somos, Jul 22 2015
a(3*n + 2) = A259896(n). - Michael Somos, Jul 22 2015

A260308 Expansion of psi(x) * phi(x^3) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jul 22 2015

nonn

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

			G.f. = 1 + x + 3*x^3 + 2*x^4 + 3*x^6 + 2*x^9 + x^10 + 2*x^12 + 4*x^13 + ...
G.f. = q + q^9 + 3*q^25 + 2*q^33 + 3*q^49 + 2*q^73 + q^81 + 2*q^97 + ...

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. A115660, A128580, A128581, A129402, A134177, A190615, A192013, A259668.

a[ n_] := If[ n < 0, 0, DivisorSum[ 8 n + 1, KroneckerSymbol[ -6, #] &]];
a[ n_] := If[ n < 0, 0, Times @@ (Which[ # <= 3, Mod[#, 2], Mod[#, 24] > 12, 1 - Mod[#2, 2], True, (#2  + 1) KroneckerSymbol[ 3, #]^#2] & @@@ FactorInteger @ (8 n + 1))];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^3] EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8)), {x, 0, n}];

{a(n) = if( n<0, 0, sumdiv( 8*n + 1, d, kronecker( -6, d)))};

{a(n) = my(A, p, e); if( n<0, 0, factor(8*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p==3, 1, p%24>12, !(e%2), (e+1) * kronecker(3, p)^e)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^6 + A)^5 / (eta(x + A) * eta(x^3 + A)^2 * eta(x^12 + A)^2), n))};

Expansion of q^(-1/8) * eta(q^2)^2 * eta(q^6)^5 / (eta(q) * eta(q^3)^2 * eta(q^12)^2) in powers of q.
Euler transform of period 12 sequence [ 1, -1, 3, -1, 1, -4, 1, -1, 3, -1, 1, -2, ...].
a(n) = A259668(2*n) = A128580(4*n) = A129402(4*n) = A134177(4*n) = A190615(4*n) = A115660(8*n + 1) = A128581(8*n + 1) = A192013(8*n + 1).

A261115 Expansion of f(x, x) * f(x^4, x^8) in powers of x where f(,) is Ramanujan's general theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Aug 08 2015

nonn

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

			G.f. = 1 + 2*x + 3*x^4 + 2*x^5 + 3*x^8 + 4*x^9 + 2*x^12 + 2*x^13 + 2*x^16 + ...
G.f. = q + 2*q^7 + 3*q^25 + 2*q^31 + 3*q^49 + 4*q^55 + 2*q^73 + 2*q^79 + ...

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. A113780, A115660, A128580, A128581, A129402, A192013, A260110.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 4, 0, x^12] / QPochhammer[ x^4, x^8], {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^8 + A) * eta(x^12 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^3 * eta(x^24 + A)), n))};

Expansion of q^(-1/6) * eta(q^2)^5 * eta(q^8) * eta(q^12)^2 / (eta(q)^2 * eta(q^4)^3 * eta(q^24)) in powers of q.
Euler transform of period 24 sequence [ 2, -3, 2, 0, 2, -3, 2, -1, 2, -3, 2, -2, 2, -3, 2, -1, 2, -3, 2, 0, 2, -3, 2, -2, ...].
a(n) = (-1)^n * A260110(n) = A128580(3*n) = A129402(3*n) = A115660(6*n + 1) = A128581(6*n + 1) = A192013(6*n + 1).
a(4*n) = A113780(n). a(4*n + 1) = 2 * A260089(n). a(4*n + 2) = a(4*n + 3) = 0.

cp's OEIS Frontend

A279541 Indices of records in A192013: Sum_{d|n} Kronecker(-6, d).

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A279542 Records in A192013: Sum_{d|n} Kronecker(-6, d).

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A115660 Expansion of (phi(q) * phi(q^6) - phi(q^2) * phi(q^3)) / 2 in powers of q where phi() is a Ramanujan theta function.

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A128581 Expansion of (phi(q^2) * phi(-q^3) - phi(-q) * phi(q^6)) / 2 in powers of q where phi() is a Ramanujan theta function.

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A263548 Expansion of f(x, x) * f(x^2, x^10) in powers of x where f(, ) is Ramanujan's general theta function.

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A035184 a(n) = Sum_{d|n} Kronecker(-1, d).

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A257920 Expansion of phi(x) * psi(x^3) in powers of x where phi(), psi() are Ramanujan theta functions.

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