A015680 -id:A015680 - cp's OEIS frontend

A330505 Expansion of e.g.f. Sum_{k>=1} arctanh(x^k).

1, 2, 8, 24, 144, 960, 5760, 40320, 524160, 4354560, 43545600, 638668800, 6706022400, 99632332800, 2092278988800, 20922789888000, 376610217984000, 9247873130496000, 128047474114560000, 2919482409811968000, 77852864261652480000

Offset: 1

Ilya Gutkovskiy, Dec 16 2019

nonn

Cf. A002131, A002448, A010050, A015128, A015680, A038048, A054785, A228274, A265024, A330504.

nmax = 21; CoefficientList[Series[Sum[ArcTanh[x^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
nmax = 21; CoefficientList[Series[-Log[EllipticTheta[4, 0, x]]/2, {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[(n - 1)! DivisorSum[n, # &, OddQ[n/#] &], {n, 1, 21}]

E.g.f.: -log(theta_4(x)) / 2.
E.g.f.: (1/2) * Sum_{k>=1} log((1 + x^k) / (1 - x^k)).
E.g.f.: log(Product_{k>=1} ((1 + x^k) / (1 - x^k))^(1/2)).
E.g.f.: Sum_{k>=1} x^(2*k - 1) / ((2*k - 1) * (1 - x^(2*k - 1))).
exp(2 * Sum_{n>=1} a(n) * x^n / n!) = g.f. of A015128.
a(n) = (n - 1)! * Sum_{d|n, n/d odd} d.

A015664 Expansion of e.g.f. theta_3^(1/2).

Original entry on oeis.org

1, 1, -1, 3, 9, -15, 135, -2205, 21105, 76545, 694575, -6392925, -56600775, 66891825, -19964169225, 741313447875, 5375639894625, 44667168170625, -2328500019470625, 5663134786183875, -466442955127524375, 11513119609487120625

Offset: 0

N. J. A. Sloane

sign

The sequence shows the coefficients of sqrt(theta_3) regarded as an exponential generating function.

			sqrt(theta_3) = 1 + q - (1/2)*q^2 + (1/2)*q^3 + (3/8)*q^4 - (1/8)*q^5 + (3/16)*q^6 - (7/16)*q^7 + (67/128)*q^8 + (27/128)*q^9 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A000122 (theta_3), A015680, A186690.

Cf. A015665, A015666, A015667, A015669, A015671, A015672, A015673, A015675, A015676, A015677, A015678, A015679.

# get basic theta series in maple
maxd:=201:
# get th2, th3, th4 = Jacobi theta constants out to degree maxd
temp0:=trunc(evalf(sqrt(maxd)))+2:
a:=0: for i from -temp0 to temp0 do a:=a+q^( (i+1/2)^2): od:
th2:=series(a,q,maxd); # A098108
a:=0: for i from -temp0 to temp0 do a:=a+q^(i^2): od:
th3:=series(a,q,maxd); # A000122
th4:=series(subs(q=-q,th3),q,maxd); # A002448
series(sqrt(th3),q,maxd); # this sequence

nmax = 25; CoefficientList[Series[EllipticTheta[3, 0, x]^(1/2), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 23 2018 *)

E.g.f. appears to equal exp( Sum_{n >= 0} x^(2*n+1)/((2*n+1)*(1 + x^(2*n+1))) ). - Peter Bala, Dec 23 2021

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} A186690(k) * a(n-k)/(n-k)!. - Seiichi Manyama, Jul 07 2023

Entry revised by N. J. A. Sloane, Oct 22 2018

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A330505 Expansion of e.g.f. Sum_{k>=1} arctanh(x^k).

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A015664 Expansion of e.g.f. theta_3^(1/2).

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