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A225956 McKay-Thompson series of class 92A for the Monster group with a(0) = 1. 2
1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 5, 5, 5, 6, 7, 8, 8, 10, 12, 12, 13, 15, 17, 18, 19, 22, 25, 27, 28, 32, 36, 38, 41, 46, 51, 54, 58, 64, 71, 76, 81, 89, 99, 105, 112, 123, 134, 143, 153, 167, 182, 194, 207, 225, 244, 260, 277, 301, 325, 346 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,9

COMMENTS

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

LINKS

G. C. Greubel, Table of n, a(n) for n = -1..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of 1/q * chi(q) * chi(q^23) in powers of q where chi() is a Ramanujan theta function.

Expansion of (eta(q^2) * eta(q^46))^2 / (eta(q) * eta(q^4) * eta(q^23) * eta(q^92)) in powers of q.

Euler transform of a period 92 sequence.

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (92 t)) = f(t) where q = exp(2 Pi i t).

G.f.: 1/x * Product_{k>0} (1 + x^(2*k - 1)) * (1 + x^(46*k - 23)).

a(n) = A112216(n) unless n=0. -(-1)^n * a(n) = A132322(n).

a(n) ~ exp(2*Pi*sqrt(n/23)) / (2 * 23^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 30 2017

EXAMPLE

G.f. = 1/q + 1 + q^2 + q^3 + q^4 + q^5 + q^6 + 2*q^7 + 2*q^8 + 2*q^9 + 2*q^10 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ 1/q QPochhammer[ -q, q^2] QPochhammer[ -q^23, q^46], {q, 0, n}];

a[ n_] := SeriesCoefficient[ 1/q Product[ 1 + q^k, {k, 1, n + 1, 2}] Product[ 1 + q^k, {k, 23, n + 1, 46}], {q, 0, n}];

PROG

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^46 + A))^2 / (eta(x + A) * eta(x^4 + A) * eta(x^23 + A) * eta(x^92 + A)), n))};

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( prod( k=1, n, 1 + k%2 * x^k, 1 + A) * prod( k=1, n\23, 1 + k%2 * x^(23*k), 1 + A), n))};

CROSSREFS

Cf. A058688, A112216, A132322.

Sequence in context: A000700 A081362 A112216 * A058688 A132322 A018118

Adjacent sequences:  A225953 A225954 A225955 * A225957 A225958 A225959

KEYWORD

nonn

AUTHOR

Michael Somos, May 21 2013

STATUS

approved

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Last modified January 24 08:16 EST 2022. Contains 350534 sequences. (Running on oeis4.)