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
Examples
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 + ...
References
- B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 153 Entry 22.
Links
- 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
Programs
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Magma
A := Basis( ModularForms( Gamma1(64), 1), 701); A[6] + A[14] + A[30] - A[35] + A[36]; /* Michael Somos, Jun 21 2015 */;
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Mathematica
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 *) -
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 */ -
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 */ -
PARI
{a(n) = if( n<0, 0, sumdiv( 8*n + 5, d, (-1)^(d\2)) / 2)}; /* Michael Somos, Jun 21 2015*/
Formula
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).
Comments