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Number of self-conjugate 4-core partitions of n.
21

%I #48 Feb 07 2024 09:01:30

%S 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,

%T 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,

%U 1,1,2,0,0,0,0,1,1,0,2,1,0,1,1,0,1,2,0

%N Number of self-conjugate 4-core partitions of n.

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

%C Also the number of positive odd solutions to equation x^2 + 4*y^2 = 8*n + 5. - _Seiichi Manyama_, May 28 2017

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

%H Seiichi Manyama, <a href="/A053692/b053692.txt">Table of n, a(n) for n = 0..10000</a>

%H F. Garvan, D. Kim and D. Stanton, <a href="http://www-users.math.umn.edu/~stant001/PAPERS/cra2.pdf">Cranks and t-cores</a>, Inventiones Math. 101 (1990) 1-17.

%H Christopher R. H. Hanusa and Rishi Nath, <a href="http://arxiv.org/abs/1201.6629">The number of self-conjugate core partitions</a>, arxiv:1201.6629 [math.NT], 2012. See Table 1 p. 15.

%H David J. Hemmer, <a href="https://arxiv.org/abs/2402.03643">Generating functions for fixed points of the Mullineux map</a>, arXiv:2402.03643 [math.CO], 2024. Table 1 p. 5 mentions this sequence.

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of psi(x) * psi(x^4) in powers of x where psi() is a Ramanujan theta function. - _Michael Somos_, Nov 03 2005

%F Expansion of chi(x) * f(-x^8)^2 in powers of x where chi(), f() are Ramanujan theta functions. - _Michael Somos_, Jul 24 2012

%F 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

%F 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

%F 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

%F Euler transform of period 8 sequence [ 1, -1, 1, 0, 1, -1, 1, -2, ...]. - _Michael Somos_, Apr 28 2003

%F 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

%F 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.

%F G.f.: Sum_{k in Z} x^k / (1 - x^(8*k + 5)). - _Michael Somos_, Nov 03 2005

%F G.f.: Sum_{k>0} -(-1)^k * x^((k^2 + k)/2) / (1 - x^(2*k - 1)). - _Michael Somos_, Jun 21 2015

%F G.f.: Product_{i>=1} (1-x^(8*i))^2*(1-x^(4*i-2))/(1-x^(2*i-1)).

%F a(9*n + 2) = a(9*n + 8) = 0. a(9*n + 5) = a(n). 2 * a(n) = A008441(2*n + 1).

%e 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 + ...

%e G.f. = q^5 + q^13 + q^29 + q^37 + q^45 + q^53 + q^61 + 2*q^85 + q^101 + q^109 + ...

%t 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 *)

%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ x^8]^2, {x, 0, n}]; (* _Michael Somos_, Jun 21 2015 *)

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^8]^2 QPochhammer[ x^2, x^4] / QPochhammer[ x, x^2], {x, 0, n}]; (* _Michael Somos_, Jun 21 2015 *)

%t 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 *)

%t a[ n_] := If[ n < 0, 0, Sum[ (-1)^Quotient[d, 2], {d, Divisors[ 8 n + 5]}] / 2]; (* _Michael Somos_, Jun 21 2015 *)

%o (PARI) {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 */

%o (PARI) {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 */

%o (PARI) {a(n) = if( n<0, 0, sumdiv( 8*n + 5, d, (-1)^(d\2)) / 2)}; /* _Michael Somos_, Jun 21 2015*/

%o (Magma) A := Basis( ModularForms( Gamma1(64), 1), 701); A[6] + A[14] + A[30] - A[35] + A[36]; /* _Michael Somos_, Jun 21 2015 */;

%Y Cf. A008441, A053693.

%K easy,nonn

%O 0,11

%A _James A. Sellers_, Feb 14 2000