A278428 -id:A278428 - cp's OEIS frontend

A081360 Expansion of q^(-1/24) (m (1-m) / 16)^(1/24) in powers of q, where m = k^2 is the parameter and q is the nome for Jacobian elliptic functions.

1, -1, 1, -2, 2, -3, 4, -5, 6, -8, 10, -12, 15, -18, 22, -27, 32, -38, 46, -54, 64, -76, 89, -104, 122, -142, 165, -192, 222, -256, 296, -340, 390, -448, 512, -585, 668, -760, 864, -982, 1113, -1260, 1426, -1610, 1816, -2048, 2304, -2590, 2910, -3264, 3658, -4097, 4582, -5120, 5718, -6378

Offset: 0

Michael Somos, Mar 18 2003

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

Number of partitions of n into distinct parts with an even number of odd parts minus partitions of n into distinct parts with an odd number of odd parts. G.f.: Product_{i=1..oo} (1+(-1)^i*x^i). - Jon Perry, Jun 04 2004

			G.f. = 1 - x + x^2 - 2*x^3 + 2*x^4 - 3*x^5 + 4*x^6 - 5*x^7 + 6*x^8 - 8*x^9 + ...
G.f. = q - q^25 + q^49 - 2*q^73 + 2*q^97 - 3*q^121 + 4*q^145 - 5*q^169 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jason Fulman, Random matrix theory over finite fields, Bull. Amer. Math. Soc. (N.S.), 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70, Eq. 2, with k=0. - _N. J. A. Sloane_, Aug 31 2014
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. 14.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
E. W. Weisstein's World of Mathematics, Elliptic Lambda Function
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol

Cf. A000009, A000700, A278428.

read theta; t1:=series(eta,q,48); t2:= q^(-1/24)*t1*subs(q=q^4,t1)/subs(q=q^2,t1)^2; series(t2,q,48); seriestolist(%); # N. J. A. Sloane, Aug 24 2007

a[ n_] := SeriesCoefficient[ 1 / QPochhammer[ -x, x^2], {x, 0, n}]; (* Michael Somos, Jul 19 2012 *)
a[ n_] := SeriesCoefficient[ 1 / Product[ 1 + x^k, {k, 1, n, 2}], {x, 0, n}]; (* Michael Somos, Jul 19 2012 *)
a[ n_] := SeriesCoefficient[ With[ {m = ModularLambda[ Log[ q] / (Pi I)]}, ( m (1 - m) / (16 q))^(1/24)], {q, 0, n}]; (* Michael Somos, Jul 19 2012 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, -x], {x, 0, n}]; (* Michael Somos, Nov 22 2016 *)
nmax = 100; CoefficientList[Series[Product[(1 + x^(2*k))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 30 2015 *)
(QPochhammer[-1, -x]/2 + O[x]^60)[[3]] (* Vladimir Reshetnikov, Nov 21 2016 *)

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

Expansion of 1 / chi(x) = chi(-x) / chi(-x^2) = f(x) / phi(x) = f(-x) / phi(-x^2) = psi(-x) / f(-x^2) = f(-x^2) / f(x) = f(-x^4) / psi(x) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of (lambda * (1 - lambda) / (16 * q))^(1/24) in powers of q where lambda is a modular elliptic function and q = exp(Pi i z) is the nome. - Michael Somos, Jul 19 2012
Expansion of q^(-1/24) * eta(q) * eta(q^4) / eta(q^2)^2 in powers of q.
Expansion of q^(-1/24) / f(t) in powers of q = exp(Pi i t) where f() is Weber's function.
Euler transform of period 4 sequence [-1, 1, -1, 0, ...].
Given g.f. A(x), B(x) = x * A(x^3)^8 satisfies 0 = f(B(x), B(x^2)) where f(u, v) = (u - v^2) * (v - u^2) - (4 * u * v * (1 - u*v))^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 16 2007
G.f.: Product_{k>0} 1 / ( 1 + x^(2k - 1)) = Product_{k>0} (1 + (-x)^k).
a(n) = (-1)^n * A000009(n). Convolution inverse of A000700.
a(n) ~ (-1)^n * exp(Pi*sqrt(n/3)) / (4*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Aug 30 2015
G.f.: (1/2)*(-1; -x)inf, where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 21 2016
G.f.: exp(-Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - Ilya Gutkovskiy, Jun 08 2018
Given g.f. A(x), B(x) = 2^(1/4) * x * A(x^24) satisfies 0 = f(B(x), B(x^5)) where f(u, v) = u^6 + v^6 + 2*u*v * ((u*v)^4 - 1). - Michael Somos, Mar 14 2019

A255526 Coefficient of x^n in Product_{k>=1} 1/(1+x^k)^n.

Original entry on oeis.org

-1, 1, -4, 17, -56, 172, -547, 1809, -6061, 20316, -68135, 229244, -774372, 2624119, -8912759, 30328593, -103382254, 352975681, -1206921212, 4132159452, -14163858895, 48601267199, -166930975524, 573872089212, -1974472043081, 6798561779868, -23425506369715

Offset: 1

Vaclav Kotesovec, Feb 24 2015

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Vaclav Kotesovec, Table of n, a(n) for n = 1..500

Cf. A008485, A270913, A278428.

Table[SeriesCoefficient[Product[1/(1+x^k)^n,{k,1,n}],{x,0,n}],{n,1,30}]
(* Calculation of constant c: *) 1/Sqrt[(4 - r^2*s^3*Derivative[0, 2][QPochhammer][-1, r*s])*Pi] /. FindRoot[{QPochhammer[-1, r*s] == 2/s, 2/s + r*s*Derivative[0, 1][QPochhammer][-1, r*s] == 0}, {r, -1/3}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Oct 03 2023 *)

a(n) ~ (-1)^n * c * d^n / sqrt(n), where d = A318204 = 3.5097543279497033404372735..., c = 0.23322106096789389697797... .

A301624 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 - x^k*A(x)^k)^k.

Original entry on oeis.org

1, -1, -1, 4, 1, -17, -6, 118, -8, -876, 625, 5966, -7486, -41937, 75969, 306312, -768637, -2164992, 7487063, 14461466, -70259884, -89410774, 646971980, 459817892, -5861484630, -1128608133, 52082250637, -15894742662, -453574650852, 366848121166, 3866670213663, -5215687717614

Offset: 0

Ilya Gutkovskiy, Mar 24 2018

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			G.f. A(x) = 1 - x - x^2 + 4*x^3 + x^4 - 17*x^5 - 6*x^6 + 118*x^7 - 8*x^8 - 876*x^9 + 625*x^10 + ...
G.f. A(x) satisfies: A(x) = (1 - x*A(x)) * (1 - x^2*A(x)^2)^2 * (1 - x^3*A(x)^3)^3 * (1 - x^4*A(x)^4)^4 * ...
log(A(x)) = -x - 3*x^2/2 + 8*x^3/3 + 13*x^4/4 - 51*x^5/5 - 120*x^6/6 + 538*x^7/7 + 781*x^8/8 - 5419*x^9/9 - 3053*x^10/10 + ... + A281267(n)*x^n/n + ...

Cf. A000219, A006195, A066398, A073592, A109085, A181315, A278428, A281267, A301455, A301456, A301625.

with(numtheory):
Order := 33:
Gser := solve(series(x*exp(add(sigma[2](n)*x^n/n, n = 1..32)), x) = y, x):
seq(coeff(Gser, y^k), k = 1..32); # Peter Bala, Feb 09 2020

From Peter Bala, Feb 09 2020: (Start)
A(x) = 1/x * series reversion of ( exp( Sum_{n >= 1} sigma_2(n)*x^n/n ) ), where sigma_2(n) = A001157(n).
Equivalently, the o.g.f. A(x) satisfies [x^n](1/A(x))^n = sigma_2(n) for n >= 1. Cf. A066398. (End)
A(x) equals (1/x) * series reversion of (x * the o.g.f. for the sequence of planar partitions A000219).  - Peter Bala, Feb 11 2020

A318204 Decimal expansion of a constant related to the asymptotics of A255526.

Original entry on oeis.org

3, 5, 0, 9, 7, 5, 4, 3, 2, 7, 9, 4, 9, 7, 0, 3, 3, 4, 0, 4, 3, 7, 2, 7, 3, 5, 2, 3, 3, 7, 5, 1, 9, 3, 6, 9, 8, 4, 5, 4, 7, 8, 9, 7, 3, 3, 9, 3, 1, 7, 3, 9, 9, 1, 1, 7, 8, 9, 8, 9, 9, 3, 7, 8, 5, 8, 5, 4, 8, 2, 1, 7, 0, 1, 5, 1, 2, 0, 0, 7, 7, 4, 4, 5, 6, 4, 8, 9, 4, 0, 8, 1, 3, 0, 7, 5, 1, 2, 1, 3, 2, 6, 4, 0, 2

Offset: 1

Vaclav Kotesovec, Aug 21 2018

			3.509754327949703340437273523375193698454789733931739911...

Cf. A255526, A278428, A303174.

RealDigits[1/r /. FindRoot[{2*r == s*QPochhammer[-1, -s], 2*r == s^2*Derivative[0, 1][QPochhammer][-1, -s]}, {r, 1/3}, {s, 1/2}, WorkingPrecision -> 120], 10, 105][[1]] (* Vaclav Kotesovec, Oct 03 2023 *)

A301625 G.f. A(x) satisfies: A(x) = Product_{k>=1} ((1 + x^kA(x)^k)/(1 - x^kA(x)^k))^k.

Original entry on oeis.org

1, 2, 10, 60, 398, 2820, 20892, 159868, 1253758, 10024070, 81400672, 669532924, 5566386324, 46701736772, 394910202608, 3362210548344, 28797181196766, 247955463799812, 2145088563952510, 18636002388075260, 162523319555310664, 1422259430668179592, 12485554521209720492, 109922263517662775292

Offset: 0

Ilya Gutkovskiy, Mar 24 2018

nonn

			G.f. A(x) = 1 + 2*x + 10*x^2 + 60*x^3 + 398*x^4 + 2820*x^5 + 20892*x^6 + 159868*x^7 + 1253758*x^8 + ...
G.f. A(x) satisfies: A(x) = ((1 + x*A(x)) * (1 + x^2*A(x)^2)^2 * (1 + x^3*A(x)^3)^3 * ...)/((1 - x*A(x)) * (1 - x^2*A(x)^2)^2 * (1 - x^3*A(x)^3)^3 * ...).
log(A(x)) = 2*x + 16*x^2/2 + 128*x^3/3 + 1056*x^4/4 + 8952*x^5/5 + 77200*x^6/6 + 673948*x^7/7 + 5937792*x^8/8 + ... + A270924(n)*x^n/n + ...

Cf. A006195, A066398, A109085, A156616, A181315, A270924, A278428, A301455, A301456, A301624.

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A081360 Expansion of q^(-1/24) (m (1-m) / 16)^(1/24) in powers of q, where m = k^2 is the parameter and q is the nome for Jacobian elliptic functions.

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A255526 Coefficient of x^n in Product_{k>=1} 1/(1+x^k)^n.

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A301624 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 - x^k*A(x)^k)^k.

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A318204 Decimal expansion of a constant related to the asymptotics of A255526.

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A301625 G.f. A(x) satisfies: A(x) = Product_{k>=1} ((1 + x^k*A(x)^k)/(1 - x^k*A(x)^k))^k.

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A301625 G.f. A(x) satisfies: A(x) = Product_{k>=1} ((1 + x^kA(x)^k)/(1 - x^kA(x)^k))^k.