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A097793 McKay-Thompson series of class 56B for the Monster group. 9
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified January 17 16:50 EST 2019. Contains 319235 sequences. (Running on oeis4.)