A112150 -id:A112150 - cp's OEIS frontend

A341245 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^6.

1, 0, 6, 6, 21, 36, 71, 132, 222, 392, 633, 1038, 1629, 2544, 3885, 5842, 8691, 12738, 18494, 26520, 37722, 53132, 74235, 102882, 141579, 193506, 262713, 354552, 475749, 634932, 842922, 1113630, 1464450, 1917254, 2499330, 3244998, 4196966, 5408004, 6943632, 8884996

Offset: 6

Ilya Gutkovskiy, Feb 07 2021

nonn

Cf. A000700, A001484, A022601, A112150, A327384, A338463, A341225, A341241, A341243, A341244, A341246, A341247, A341251.

g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 6):
seq(a(n), n=6..45);  # Alois P. Heinz, Feb 07 2021

nmax = 45; CoefficientList[Series[(-1 + Product[1/(1 + (-x)^k), {k, 1, nmax}])^6, {x, 0, nmax}], x] // Drop[#, 6] &

G.f.: (-1 + Product_{k>=1} (1 + x^(2*k - 1)))^6.

A022601 Expansion of Product_{m>=1} (1+q^m)^(-6).

Original entry on oeis.org

1, -6, 15, -26, 51, -102, 172, -276, 453, -728, 1128, -1698, 2539, -3780, 5505, -7882, 11238, -15918, 22259, -30810, 42438, -58110, 78909, -106392, 142770, -190698, 253179, -334266, 439581, -575784, 750613, -974316, 1260336, -1624702, 2086530, -2670162

Offset: 0

N. J. A. Sloane

sign

McKay-Thompson series of class 8F for the Monster group.

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

			G.f. = 1 - 6*x + 15*x^2 - 26*x^3 + 51*x^4 - 102*x^5 + 172*x^6 - 276*x^7 + ...
T8F = 1/q - 6*q^3 + 15*q^7 - 26*q^11 + 51*q^15 - 102*q^19 + 172*q^23 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for McKay-Thompson series for Monster simple group

Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
Cf. A022571, A058088, A081362, A112145, A112150.
Column k=6 of A286352.

a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^6, {x, 0, n}]; (* Michael Somos, Jul 01 2014 *)
nmax = 50; CoefficientList[Series[Product[1/(1 + x^k)^6, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^2 + A))^6, n))}; /* Michael Somos, Jul 01 2014 */

Expansion of chi(-x)^6 in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Jul 01 2014
Expansion of q^(1/4) * 2 * k'(q) / k(q)^(1/2) in powers of q where k() is the elliptic modulus. - Michael Somos, Jul 01 2014
Expansion of q^(1/4) * (eta(q) / eta(q^2))^6 in powers of q. - Michael Somos, Jul 01 2014
Euler transform of period 2 sequence [ -6, 0, ...]. - Michael Somos, Jul 01 2014
Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u - v^3) * (u^3 - v) - 3*u*v * (21 + 6*u*v). - Michael Somos, Jul 01 2014
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 8 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A022571. - Michael Somos, Jul 01 2014
Convolution inverse of A022571. Convolution sixth power of A081362. - Michael Somos, Jul 01 2014
a(n) = (-1)^n * A112150(n) = A058088(2*n) = A112145(2*n). - Michael Somos, Jul 01 2014
a(n) ~ (-1)^n * exp(Pi*sqrt(n)) / (2^(3/2) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
a(0) = 1, a(n) = -(6/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 05 2017
G.f.: exp(-6*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

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A341245 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^6.

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A022601 Expansion of Product_{m>=1} (1+q^m)^(-6).

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