1,2

Coefficients of the cusp form of weight 12 for the full modular group.

It is conjectured that tau(n) is never zero.

Number of partitions of n into an even number of distinct parts - partitions of n into an odd number of distinct parts, with 24 types of each part. - Jon Perry (perry(AT)globalnet.co.uk), Apr 04 2004

M. J. Hopkins mentions that the only known primes p for which tau(p) == 1 mod p are 11, 23 and 691, that it is an open problem to decide if there are infinitely many such p and that no others are known below 35000. Simon Plouffe has now searched up to tau(314747) and found no other examples. - N. J. A. Sloane (njas(AT)research.att.com), Mar 25 2007

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 77, Eq. (32.2).

M. J. Hopkins, Algebraic toplogy and modular forms, Proc. Internat. Congress Math., Beijing 2002, Vol. I, pp. 291-317.

Bruce Jordan and Blair Kelly (blair.kelly(AT)att.net), The vanishing of the Ramanujan tau function, preprint, 2001.

M. Kaneko and D. Zagier, Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials, pp. 97-126 of D. A. Buell and J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, Amer. Math. Soc., 1998.

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B. C. Berndt and K. Ono, Ramanujan's unpublished manuscript...

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Index entries for "core" sequences

Index entries for expansions of Product_{k >= 1} (1-x^k)^m

G.f.: x Product_{k=1..infinity} (1 - x^k)^24.

|a(n)| = O(n^(11/2 + epsilon)), |a(p)| <= 2 p^(11/2) if p is prime. These were conjectured by Ramanujan and proved by Deligne.

Zagier says: The proof of these formulae, if written out from scratch, has been estimated at 2000 pages; in his book Manin cites this as a probable record for the ratio: `length of proof:length of statement' in the whole of mathematics.

G.f. A(x) satisfies 0=f(A(x),A(x^2),A(x^4)) where f(u,v,w)=uw(u+48v+4096w)-v^3. - Michael Somos, Jul 19 2004

x Product (1 - x^k)^24 = x - 24*x^2 + 252*x^3 - 1472*x^4 + 4830*x^5 - 6048*x^6 - 16744*x^7 + 84480*x^8 - 113643*x^9 + ...

M := 50; t1 := series(x*mul((1-x^k)^24, k=1..M), x, M); A000594 := n-> coeff(t1, x, n);

CoefficientList[ Take[ Expand[ Product[ (1 - x^k)^24, {k, 1, 30} ]], 30], x] (* Or *)

(* first do *) Needs["NumberTheory`Ramanujan`"] (* then *) Table[ RamanujanTau[n], {n, 30}] (from Dean Hickerson (dean.hickerson(AT)yahoo.com), Jan 03 2003)

(MAGMA) M12:=ModularForms(Gamma0(1), 12); t1:=Basis(M12)[2]; PowerSeries(t1[1], 100); Coefficients($1);

(PARI) a(n)=if(n<1, 0, polcoeff(x*eta(x+x*O(x^n))^24, n))

(PARI) a(n)=if(n<1, 0, polcoeff(x*(sum(i=1, (sqrtint(8*n-7)+1)\2, (-1)^i*(2*i-1)*x^((i^2-i)/2), O(x^n)))^8, n))

Cf. A076847 (tau(p)), A037955, A027364, A037945, A037946, A037947, A008408 (Leech).

For a(n) mod N for various values of N see A046694, A126811-...

Sequence in context: A052652 A052732 A086603 this_sequence A022716 A051828 A076847

Adjacent sequences: A000591 A000592 A000593 this_sequence A000595 A000596 A000597

sign,easy,core,mult,nice

N. J. A. Sloane (njas(AT)research.att.com).