A246950 -id:A246950 - cp's OEIS frontend

A053692 Number of self-conjugate 4-core partitions of n.

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

Offset: 0

James Sellers, Feb 14 2000

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 + 4*y^2 = 8*n + 5. - Seiichi Manyama, May 28 2017

			G.f. = 1 + x + x^3 + x^4 + x^5 + x^6 + x^7 + 2*x^10 + x^12 + x^13 + x^14 + 2*x^15 + ...
G.f. = q^5 + q^13 + q^29 + q^37 + q^45 + q^53 + q^61 + 2*q^85 + q^101 + q^109 + ...

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 153 Entry 22.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
F. Garvan, D. Kim and D. Stanton, Cranks and t-cores, Inventiones Math. 101 (1990) 1-17.
Christopher R. H. Hanusa and Rishi Nath, The number of self-conjugate core partitions, arxiv:1201.6629 [math.NT], 2012. See Table 1 p. 15.
David J. Hemmer, Generating functions for fixed points of the Mullineux map, arXiv:2402.03643 [math.CO], 2024. Table 1 p. 5 mentions this sequence.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A008441, A053693.

A := Basis( ModularForms( Gamma1(64), 1), 701); A[6] + A[14] + A[30] - A[35] + A[36]; /* Michael Somos, Jun 21 2015 */;

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x (1/2)] EllipticTheta[ 2, 0, x^2] / (4 x^(5/8)), {x, 0, n}]; (* Michael Somos, Jun 21 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ x^8]^2, {x, 0, n}]; (* Michael Somos, Jun 21 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^8]^2 QPochhammer[ x^2, x^4] / QPochhammer[ x, x^2], {x, 0, n}]; (* Michael Somos, Jun 21 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q^2]^2 - 2 EllipticTheta[ 2, Pi/4, q^2]^2) / 16, {q, 0, 8 n + 5}]; (* Michael Somos, Jun 21 2015 *)
a[ n_] := If[ n < 0, 0, Sum[ (-1)^Quotient[d, 2], {d, Divisors[ 8 n + 5]}] / 2]; (* Michael Somos, Jun 21 2015 *)

{a(n) = my(A); if( n<0, 0, A = sum( k=0, ceil( sqrtint(8*n + 1)/2), x^((k^2 + k)/2), x * O(x^n)); polcoeff( A * subst(A + x * O(x^(n\4)), x, x^4), n))}; /* Michael Somos, Nov 03 2005 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^8 + A)^2 / (eta(x + A) * eta(x^4 + A)), n))}; /* Michael Somos, Apr 28 2003 */

{a(n) = if( n<0, 0, sumdiv( 8*n + 5, d, (-1)^(d\2)) / 2)}; /* Michael Somos, Jun 21 2015*/

Expansion of psi(x) * psi(x^4) in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Nov 03 2005
Expansion of chi(x) * f(-x^8)^2 in powers of x where chi(), f() are Ramanujan theta functions. - Michael Somos, Jul 24 2012
Expansion of f(x, x^7) * f(x^3, x^5) = f(x, x^3) * f(x^4, x^12) in powers of x where f(,) is the Ramanujan general theta function. - Michael Somos, Jun 21 2015
Expansion of (psi(x)^2 - psi(-x)^2) / (4*x) in powers of x^2 where psi() is a Ramanujan theta function. - Michael Somos, Jun 21 2015
Expansion of q^(-5/8) * eta(q^2)^2 * eta(q^8)^2 / (eta(q) * eta(q^4)) in powers of q. - Michael Somos, Apr 28 2003
Euler transform of period 8 sequence [ 1, -1, 1, 0, 1, -1, 1, -2, ...]. - Michael Somos, Apr 28 2003
a(n) = 1/2 * b(8*n + 5), where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4), b(p^e) = e+1 if p == 1 (mod 4). - Michael Somos, Jul 24 2012
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246950.
G.f.: Sum_{k in Z} x^k / (1 - x^(8*k + 5)). - Michael Somos, Nov 03 2005
G.f.: Sum_{k>0} -(-1)^k * x^((k^2 + k)/2) / (1 - x^(2*k - 1)). - Michael Somos, Jun 21 2015
G.f.: Product_{i>=1} (1-x^(8*i))^2*(1-x^(4*i-2))/(1-x^(2*i-1)).
a(9*n + 2) = a(9*n + 8) = 0. a(9*n + 5) = a(n). 2 * a(n) = A008441(2*n + 1).

A104794 Expansion of theta_4(q)^2 in powers of q.

Original entry on oeis.org

1, -4, 4, 0, 4, -8, 0, 0, 4, -4, 8, 0, 0, -8, 0, 0, 4, -8, 4, 0, 8, 0, 0, 0, 0, -12, 8, 0, 0, -8, 0, 0, 4, 0, 8, 0, 4, -8, 0, 0, 8, -8, 0, 0, 0, -8, 0, 0, 0, -4, 12, 0, 8, -8, 0, 0, 0, 0, 8, 0, 0, -8, 0, 0, 4, -16, 0, 0, 8, 0, 0, 0, 4, -8, 8, 0, 0, 0, 0, 0, 8

Offset: 0

Michael Somos, Mar 26 2005

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Quadratic AGM theta functions: a(q) (see A004018), b(q) (A104794), c(q) (A005883).
In the Arithmetic-Geometric Mean, if a = theta_3(q)^2, b = theta_4(q)^2 then a' := (a+b)/2 = theta_3(q^2)^2, b' := sqrt(a*b) = theta_4(q^2)^2.

			G.f. = 1 - 4*q + 4*q^2 + 4*q^4 - 8*q^5 + 4*q^8 - 4*q^9 + 8*q^10 - 8*q^13 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 576.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000203, A000729, A001934, A002131, A004018, A008441, A053692, A113407, A113652, A122856, A122865, A204531, A227695, A246950, A256014, A258210.

# JacobiTheta4 is defined in A002448.
A104794List(len) = JacobiTheta4(len, 2)
A104794List(102) |> println # Peter Luschny, Mar 12 2018

A := Basis( ModularForms( Gamma1(8), 1), 100); A[1] - 4*A[2] + 4*A[3]; /* Michael Somos, Jan 31 2015 */

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^2, {q, 0, n}];
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ Sqrt[1 - m] EllipticK[m] / (Pi/2), {q, 0, n}]];
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m)^(1/4) EllipticK[m] / (Pi/2), {q, 0, 2 n}]];
a[ n_] := With[ {m = InverseEllipticNomeQ @ -q}, SeriesCoefficient[ EllipticK[ m] / (Pi/2), {q, 0, n}]]; (* Michael Somos, Jun 06 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], (-1)^n 4 DivisorSum[ n, KroneckerSymbol[ -4, #] &]]; (* Michael Somos, Jun 06 2015 *)

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

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

{a(n) = if( n<0, 0, polcoeff( 1 + 4 * sum( k=1, n, (-x)^k / (1 + x^(2*k)), x * O(x^n)), n))};

Expansion of phi(-q)^2 = 2 * phi(q^2)^2 - phi(q)^2 = (phi(q) - 2*phi(q^4))^2 = f(-q)^3 / psi(q) = phi(-q^2)^4 / phi(q)^2 = psi(-q)^4 / psi(q^2)^2 = psi(q)^2 * chi(-q)^6 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of (1-k^2)^(1/2) K(k^2) / (Pi/2) in powers of q where q is Jacobi's nome, k is the elliptic modulus and K() is the complete elliptic integral of the first kind.
Expansion of  K(k^2) / (Pi/2) in powers of -q where q is Jacobi's nome, k is the elliptic modulus and K() is the complete elliptic integral of the first kind. - Michael Somos, Jun 08 2015
Expansion of eta(q)^4 / eta(q^2)^2 in powers of q.
Euler transform of period 2 sequence [ -4, -2, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v * (u^2 + v^2) - 2*u*w^2.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^2 - 2*u1*u3 + 4*u2*u6 - 3*u3^2.
Moebius transform is period 8 sequence [ -4, 8, 4, 0, -4, -8, 4, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 16 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A008441.
G.f.: theta_4(q)^2 = (Sum_{k in Z} (-q)^(k^2))^2 = (Product_{k>0} (1 - q^(2*k)) * (1 - q^(2*k - 1))^2)^2.
G.f.: 1 + 4 * Sum_{k>0} (-x)^k / (1 + x^(2*k)). - Michael Somos, Jun 08 2015
a(4*n + 3) = 0. a(n) = (-1)^n * A004018(n) = a(2*n). a(4*n + 1) = -4 * A008441(n). a(n) = -4 * A113652(n) unless n=0. a(6*n + 2) = 4 * A122865(n). a(6*n + 4) = 4 * A122856(n). a(8*n + 1) = -4 * A113407(n). a(8*n + 5) = -8 * A053692(n).
a(n) = a(9*n) = A204531(8*n) = A246950(8*n) = A256014(9*n) = A258210(n). - Michael Somos, Jun 08 2015
Convolution inverse of A001934. Convolution with A000729 is A227695. - Michael Somos, Jun 08 2015
G.f.: 2 * Sum_{k in Z} (-1)^k * x^(k*(k + 1)/2) / (1 + x^k). - Michael Somos, Nov 05 2015
a(0) = 1, a(n) = -(4/n)*Sum_{k=1..n} A002131(k)*a(n-k) for n > 0. - Seiichi Manyama, May 02 2017
G.f.: exp(2*Sum_{k>=1} (sigma(k) - sigma(2*k))*x^k/k). - Ilya Gutkovskiy, Sep 19 2018

A134343 Expansion of psi(-x)^2 in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Oct 21 2007

sign

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

Number 57 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = 1 - 2*x + x^2 - 2*x^3 + 2*x^4 + 3*x^6 - 2*x^7 - 2*x^9 + 2*x^10 + ...
G.f. = q - 2*q^5 + q^9 - 2*q^13 + 2*q^17 + 3*q^25 - 2*q^29 - 2*q^37 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See p. 44.
Yves 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. A008441, A053692, A113407, A121613, A143379, A209942, A213791, A246862, A246683, A246950, A259285, A259287.

A := Basis( ModularForms( Gamma1(64), 1), 321); A[2] - 2*A[6] + A[10] - 2*A[14] + 2*A[18] + 3*A[26] - 2*A[30] + 2*A[35] - 2*A[36]; /* Michael Somos, Jun 22 2015 */;

a[ n_] := If[ n < 0, 0, (-1)^n DivisorSum[ 4 n + 1, (-1)^Quotient[#, 2] &]]; (* Michael Somos, Jun 22 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, x^(1/2)]^2 / (2 x^(1/4)), {x, 0, n}]; (* Michael Somos, Jun 22 2015 *)
a[ n_] := SeriesCoefficient[(QPochhammer[ x] QPochhammer[ x^4] / QPochhammer[ x^2])^2, {x, 0, n}]; (* Michael Somos, Jun 22 2015 *)

{a(n) = if( n<0, 0, (-1)^n * sumdiv( 4*n + 1, d, (-1)^(d\2)))};

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

Expansion of q^(-1/4) * (eta(q) * eta(q^4) / eta(q^2))^2 in powers of q.
Euler transform of period 4 sequence [ -2, 0, -2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 8 (t/i) f(t) where q = exp(2 Pi i t).
a(n) = b(4*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4), b(p^e) = e+1 if p == 1 (mod 8), b(p^e) = (-1)^e * (e+1) if p == 5 (mod 8).
G.f.: (Product_{k>0} (1 - x^k) * (1 + x^(2*k)))^2.
a(9*n + 5) = a(9*n + 8) = 0. a(n) = (-1)^n * A008441(n). a(2*n) = A113407(n). a(2*n + 1) = -2 * A053692(n).
2 * a(n) = A204531(4*n + 1) = - A246950(n). a(4*n) = A246862(n). a(4*n + 2) = A246683(n). - Michael Somos, Jun 22 2015
a(4*n + 1) = -2 * A259287(n). a(4*n + 3) = -2 * A259285(n). - Michael Somos, Jun 24 2015
Convolution square is A121613. Convolution cube is A213791. Convolution with A000009 is A143379. Convolution with A000143 is A209942. Michael Somos, Jun 22 2015
G.f.: Sum_{k>0 odd} (x^k + x^(3*k)) / (1 + x^(4*k)) * (-1)^floor((k+1) / 4). - Michael Somos, Jun 22 2015
G.f.: Sum_{k>0 odd} (x^k - x^(3*k)) / (1 + x^(4*k)) * (-1)^floor(k / 4). - Michael Somos, Jun 22 2015

A204531 Expansion of phi(q) * phi(-q^4) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jan 15 2012

sign

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

			G.f. = 1 + 2*q - 4*q^5 - 4*q^8 + 2*q^9 - 4*q^13 + 4*q^16 + 4*q^17 + ...

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. A053692, A104794, A113407, A134343, A246950.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q^4], {q, 0, n}]; (* Michael Somos, Jun 10 2015 *)

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

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); 2 * (-1)^(n%8==5) * prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 2 * (e>2) * (-1)^(e<4), p%4==1, e+1, !(e%2))))};

Expansion of eta(q^2)^5 / (eta(q)^2 * eta(q^8)) in powers of q.
Euler transform of period 8 sequence [ 2, -3, 2, -3, 2, -3, 2, -2, ...].
G.f.: Product_{k>0} (1 - x^(2*k))^5 / ((1 - x^k)^2 * (1 - x^(8*k))).
a(4*n + 2) = a(4*n + 3) = a(8*n + 4) = 0. a(8*n) = A104794(n). a(4*n + 1) = 2 * A134343(n).
a(n) = (-1)^n * A246950(n). a(8*n + 1) = 2 * A113407(n). a(8*n + 5) = -4 * A053692(n). - Michael Somos, Jun 10 2015

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A053692 Number of self-conjugate 4-core partitions of n.

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A104794 Expansion of theta_4(q)^2 in powers of q.

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A134343 Expansion of psi(-x)^2 in powers of x where psi() is a Ramanujan theta function.

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A204531 Expansion of phi(q) * phi(-q^4) in powers of q where phi() is a Ramanujan theta function.

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