A096563 McKay-Thompson series of class 25a for the Monster group.

1, 0, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, -1, 0, 0, 1, 0, 1, 0, 0, -1, 0, -1, 0, 0, -2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, -2, 0, 0, 2, 0, 3, 0, 0, -1, 0, -2, 0, 0, -3, 0, 0, 0, 0, -1, 0, 2, 0, 0, 3, 0, -4, 0, 0, 3, 0, 4, 0, 0, -2, 0, -3, 0, 0, -5, 0, 1, 0, 0, -1, 0, 3, 0, 0, 6, 0, -6, 0, 0

Offset

-1,41

Links

Seiichi Manyama, Table of n, a(n) for n = -1..1000

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).

T. Horie and N. Kanou, Certain modular functions similar to the Dedekind eta function, Abh. Math. Sem. Univ. Hamburg 72 (2002), 89-117. MR1941549 (2003j:11043)

Index entries for McKay-Thompson series for Monster simple group

Formula

Expansion of 1 + eta(q) / eta(q^25) in powers of q.

G.f.: 1 + x^-1 * (Prod_{k>0} (1 - x^k) / (1 - x^(25*k))).

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u^2 - v)* (u - v^2) - 2*(u - 1)^2 - 2*(v - 1)^2.

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 + u*w + w^2 - v - v^2*(u + w) - 2*(u + w) + 4.

a(n) = A096562(n) unless n=0.

Example

T25a = 1/q - q + q^4 + q^6 - q^11 - q^14 + q^21 + q^24 - q^26 + q^29 + q^31 + ...

Mathematica

a[ n_] := With[ {m = n + 1}, SeriesCoefficient[ q + Product[ 1 - q^k, {k, m}] / Product[ 1 - q^k, {k, 25, m, 25}], {q, 0, m}]];
QP = QPochhammer; s = q + QP[q]/QP[q^25] + O[q]^110; CoefficientList[s, q] (* Jean-François Alcover, Nov 15 2015 *)

Prog

(PARI) {a(n) = my(A, m); if( n<-1, 0, m=5; A = x + O(x^6); while( m < n+2, m*=5; A = x * subst( (A * (1 - 2*A + 4*A^2 - 3*A^3 + A^4) / (1 + 3*A + 4*A^2 + 2*A^3 + A^4) / x)^(1/5), x, x^5)); polcoeff( 1/A - A, n))}
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( x + eta(x + A)/ eta(x^25 + A), n))}

Crossrefs

Cf. A092885, A096562.

Sequence in context: A324179 A216577 A096562 * A216512 A078359 A107329

Adjacent sequences: A096560 A096561 A096562 * A096564 A096565 A096566

Keyword

sign

Author

Michael Somos, Jul 02 2004

Status

approved