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 A053692 Number of self-conjugate 4-core partitions of n. 21
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,11 COMMENTS 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 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. Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions 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). EXAMPLE 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 + ... 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 *) PROG (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*/ (Magma) A := Basis( ModularForms( Gamma1(64), 1), 701); A[6] + A[14] + A[30] - A[35] + A[36]; /* Michael Somos, Jun 21 2015 */; CROSSREFS Cf. A008441, A053693. Sequence in context: A272903 A321458 A226194 * A341024 A286934 A282714 Adjacent sequences: A053689 A053690 A053691 * A053693 A053694 A053695 KEYWORD easy,nonn AUTHOR James A. Sellers, Feb 14 2000 STATUS approved

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Last modified December 2 07:49 EST 2023. Contains 367510 sequences. (Running on oeis4.)