A097793 McKay-Thompson series of class 56B for the Monster group.

1, 1, 1, 2, 2, 3, 4, 4, 5, 7, 8, 10, 12, 14, 17, 21, 24, 28, 34, 39, 46, 53, 61, 71, 82, 94, 108, 124, 142, 162, 185, 210, 238, 271, 306, 345, 390, 439, 494, 556, 623, 698, 783, 875, 977, 1092, 1216, 1354, 1508, 1674, 1859, 2064, 2286, 2532, 2803, 3098, 3424

Offset

0,4

Comments

Number of partitions of n into distinct parts not divisible by 7.

Also McKay-Thompson series of class 56C for Monster. - Michel Marcus, Feb 19 2014

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

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

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 12.

Index entries for McKay-Thompson series for Monster simple group

Formula

Euler transform of period 14 sequence [ 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, ...].

Expansion of q^(1/4) * eta(q^2) * eta(q^7) / (eta(q) * eta(q^14)) in powers of q.

G.f.: Product_{k>0} (1 + x^k) / (1 + x^(7*k)).

a(n) ~ exp(Pi*sqrt(2*n/7)) / (2 * 14^(1/4) * n^(3/4)) * (1 - (3*sqrt(7)/ (8*Pi*sqrt(2)) + Pi/(4*sqrt(14))) / sqrt(n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 21 2017

Example

1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 4*x^7 + 5*x^8 + 7*x^9 + 8*x^10 +...
T56B = 1/q + q^3 + q^7 + 2q^11 + 2q^15 + 3q^19 + 4q^23 + 4q^27 +...

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 + x^k) / (1 + x^(7*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)
QP = QPochhammer; s = QP[q^2]*(QP[q^7]/(QP[q]*QP[q^14])) + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 12 2015 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( prod( k=1, n, 1 + x^k, 1 + A) / prod( k=1, n\7, 1 + x^(7*k), 1 + A), n))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^7 + A) / (eta(x + A) * eta(x^14 + A)), n))}

Crossrefs

Cf. A113297.

Cf. A000700 (m=2), A003105 (m=3), A070048 (m=4), A096938 (m=5), A261770 (m=6), A261771 (m=8), A112193 (m=9), A261772 (m=10).

Sequence in context: A112582 A104648 A141271 * A015742 A015754 A207613

Adjacent sequences: A097790 A097791 A097792 * A097794 A097795 A097796

Keyword

nonn

Author

Michael Somos, Aug 24 2004

Status

approved