A022601 -id:A022601 - 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.

A339721 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(-s))^6.

Original entry on oeis.org

1, -6, -6, 15, -6, 30, -6, -26, 15, 30, -6, -60, -6, 30, 30, 51, -6, -60, -6, -60, 30, 30, -6, 96, 15, 30, -26, -60, -6, -114, -6, -102, 30, 30, 30, 96, -6, 30, 30, 96, -6, -114, -6, -60, -60, 30, -6, -210, 15, -60, 30, -60, -6, 96, 30, 96, 30, 30, -6, 156, -6, 30, -60, 172, 30

Offset: 1

Ilya Gutkovskiy, Dec 14 2020

sign

Cf. A022601, A316441, A339338, A339717, A339718, A339719, A339720, A339722.

a(1) = 1; a(n) = -Sum_{d|n, d < n} A339338(n/d) * a(d).

a(p^k) = A022601(k) for prime p.

A022600 Expansion of Product_{m>=1} (1+q^m)^(-5).

Original entry on oeis.org

1, -5, 10, -15, 30, -56, 85, -130, 205, -315, 465, -665, 960, -1380, 1925, -2651, 3660, -5020, 6775, -9070, 12126, -16115, 21220, -27765, 36235, -47101, 60810, -78115, 100105, -127825, 162391, -205530, 259475, -326565

Offset: 0

N. J. A. Sloane

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. Related to Expansion of Product_{m>=1} (1+q^m)^k: A022627 (k=-32), A022626 (k=-31), A022625 (k=-30), A022624 (k=-29), A022623 (k=-28), A022622 (k=-27), A022621 (k=-26), A022620 (k=-25), A007191 (k=-24), A022618 (k=-23), A022617 (k=-22), A022616 (k=-21), A022615 (k=-20), A022614 (k=-19), A022613 (k=-18), A022612 (k=-17), A022611 (k=-16), A022610 (k=-15), A022609 (k=-14), A022608 (k=-13), A007249 (k=-12), A022606 (k=-11), A022605 (k=-10), A022604 (k=-9), A007259 (k=-8), A022602 (k=-7), A022601 (k=-6), this sequence (k=-5), A022599 (k=-4), A022598 (k=-3), A022597 (k=-2), A081362 (k=-1), A000009 (k=1), A022567 (k=2), A022568 (k=3), A022569 (k=4), A022570 (k=5), A022571 (k=6), A022572 (k=7), A022573 (k=8), A022574 (k=9), A022575 (k=10), A022576 (k=11), A022577 (k=12), A022578 (k=13), A022579 (k=14), A022580 (k=15), A022581 (k=16), A022582 (k=17), A022583 (k=18), A022584 (k=19), A022585 (k=20), A022586 (k=21), A022587 (k=22), A022588 (k=23), A014103 (k=24), A022589 (k=25), A022590 (k=26), A022591 (k=27), A022592 (k=28), A022593 (k=29), A022594 (k=30), A022595 (k=31), A022596 (k=32), A025233 (k=48).

Column k=5 of A286352.

nmax = 50; CoefficientList[Series[Product[1/(1 + x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)

x='x+O('x^50); Vec(prod(m=1, 50, (1 + x^m)^(-5))) \\ Indranil Ghosh, Apr 05 2017

a(n) ~ (-1)^n * 5^(1/4) * exp(Pi*sqrt(5*n/6)) / (2^(7/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
a(0) = 1, a(n) = -(5/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 05 2017
G.f.: exp(-5*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

A112150 McKay-Thompson series of class 16a for the Monster group.

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, 3406695, 4333590

Offset: 0

Michael Somos, Aug 28 2005

nonn

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 + ...
T16a = 1/q + 6*q^3 + 15*q^7 + 26*q^11 + 51*q^15 + 102*q^19 + 172*x^23 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for McKay-Thompson series for Monster simple group
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Comm. Algebra 22, No. 13, 5175-5193 (1994).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A022601, A107635, A112142.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2]^6, {x, 0, n}]; (* Michael Somos, Jul 03 2014 *)
nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k+1))^6, {k, 0, 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^2 + A)^2 / (eta(x + A) * eta(x^4 + A)))^6, n))}; /* Michael Somos, Jul 03 2014 */

Expansion of chi(x)^6 in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Jul 03 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 03 2014
Expansion of q^(1/4) * (eta(q^2)^2 / (eta(q) * eta(q^4)))^6 in powers of q. - Michael Somos, Jul 03 2014
Euler transform of period 4 sequence [ 6, -6, 6, 0, ...]. - Michael Somos, Jul 03 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) = (v^3 - u) * (u^3 - v) - 9*u*v * (-7 + 2*u*v). -  Michael Somos, Jul 03 2014
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) =  f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 03 2014
G.f.: Product_{k>0} (1 + (-x)^k)^-6 = Product_{k>0} (1 + x^(2*k - 1))^6. - Michael Somos, Jul 03 2014
Convolution square is A112142. Convolution square of A107635. - Michael Somos, Jul 03 2014
a(n) = (-1)^n * A022601(n). - Michael Somos, Jul 03 2014
a(n) ~ exp(Pi*sqrt(n)) / (2^(3/2) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
G.f.: exp(6*Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - Ilya Gutkovskiy, Jun 07 2018

cp's OEIS Frontend

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

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A339721 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(-s))^6.

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A022600 Expansion of Product_{m>=1} (1+q^m)^(-5).

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A112150 McKay-Thompson series of class 16a for the Monster group.

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