A002318 -id:A002318 - cp's OEIS frontend

A001934 Expansion of 1/theta_4(q)^2 in powers of q.

1, 4, 12, 32, 76, 168, 352, 704, 1356, 2532, 4600, 8160, 14176, 24168, 40512, 66880, 108876, 174984, 277932, 436640, 679032, 1046016, 1597088, 2418240, 3632992, 5417708, 8022840, 11802176, 17252928, 25070568, 36223424, 52053760, 74414412

Offset: 0

N. J. A. Sloane, Simon Plouffe

Euler transform of period 2 sequence [ 4, 2, ...].
The Cayley reference actually is to A004403. - Michael Somos, Feb 24 2011
Number of overpartition pairs, see Lovejoy reference. - _Joerg Arndt, Apr 03 2011
In general, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^m and m>=1, then a(n) ~ exp(Pi*sqrt(m*n)) * m^((m+1)/4) / (2^(3*(m+1)/2) * n^((m+3)/4)). - Vaclav Kotesovec, Aug 17 2015

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129]
B. Kim, Overpartition pairs modulo powers of 2, Discrete Math., 311 (2011), 835-840.
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. 8.
Jeremy Lovejoy, Overpartition pairs, Annales de l'institut Fourier, vol.56, no.3, p.781-794, 2006.

Cf. A000122, A000203, A004403, A002318, A002513, A015128, A004404-A004425.

# JacobiTheta4 is defined in A002448.
A001934List(len) = JacobiTheta4(len, -2)
A001934List(33) |> println # Peter Luschny, Mar 12 2018

Maple
```
mul((1+x^n)^2/(1-x^n)^2,n=1..256);
```

CoefficientList[Series[1/EllipticTheta[4, 0, q]^2, {q, 0, 32}], q]  (* Jean-François Alcover, Jul 18 2011 *)
nmax = 40; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 17 2015 *)
QP = QPochhammer; s = QP[q^2]^2/QP[q]^4 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Dec 01 2015, adapted from PARI *)

my(N=33, x='x+O('x^N)); Vec(prod(i=1, N, (1+x^i)^2/(1-x^i)^2))

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 / eta(x + A)^4, n))} /* Michael Somos, Feb 09 2006 */

G.f.: Product ( 1 - x^k )^{-c(k)}, c(k) = 4, 2, 4, 2, 4, 2, ....
G.f.: Product{i>=1} (1+x^i)^2/(1-x^i)^2. - Jon Perry, Apr 04 2004
Expansion of eta(q^2)^2/eta(q)^4 in powers of q, where eta(x)=prod(n>=1,1-q^n).
a(n) = (-1)^n * A004403(n). a(n) = 4 * A002318(n) unless n=0. - Michael Somos, Feb 24 2011
a(n) ~ exp(Pi*sqrt(2*n)) / (2^(15/4) * n^(5/4)) * (1 - 15/(8*Pi*sqrt(2*n)) + 105/(256*Pi^2*n)). - Vaclav Kotesovec, Aug 17 2015, extended Jan 22 2017
a(0) = 1, a(n) = (4/n)*Sum_{k=1..n} A002131(k)*a(n-k) for n > 0. - Seiichi Manyama, May 02 2017
G.f.: exp(2*Sum_{k>=1} (sigma(2*k) - sigma(k))*x^k/k). - Ilya Gutkovskiy, Sep 19 2018
The g.f. A(q^2) = 1/(F(q)*F(-q)), where F(q) = theta_3(q) = Sum_{n = -oo..oo} q^(n^2) is the g.f. of A000122. Cf. A002513. - Peter Bala, Sep 26 2023

More terms from James Sellers, Sep 08 2000

Edited by N. J. A. Sloane, May 13 2008 to remove an incorrect g.f.

A216273 Triangle generated by Sum_{n>=1, k=1..n} T(n,k)x^ny^k/n = log(1 + Sum_{n>=1} y*x^(n^2)), where coefficients are read by rows.

Original entry on oeis.org

1, 0, -1, 0, 0, 1, 4, 0, 0, -1, 0, -5, 0, 0, 1, 0, 0, 6, 0, 0, -1, 0, 0, 0, -7, 0, 0, 1, 0, -4, 0, 0, 8, 0, 0, -1, 9, 0, 9, 0, 0, -9, 0, 0, 1, 0, -10, 0, -15, 0, 0, 10, 0, 0, -1, 0, 0, 11, 0, 22, 0, 0, -11, 0, 0, 1, 0, 0, 4, -12, 0, -30, 0, 0, 12, 0, 0, -1, 0, -13, 0, -13, 13, 0, 39, 0, 0, -13, 0, 0, 1, 0, 0, 28, 0, 28, -14, 0, -49, 0, 0, 14, 0, 0, -1

Offset: 1

Paul D. Hanna, Mar 16 2013

			G.f.: A(x,y) = y*x - y^2*x^2/2 + y^3*x^3/3 + (-y^4 + 4*y)*x^4/4 + (y^5 - 5*y^2)*x^5/5 + (-y^6 + 6*y^3)*x^6/6 + (y^7 - 7*y^4)*x^7/7 + (-y^8 + 8
*y^5 - 4*y^2)*x^8/8 + (y^9 - 9*y^6 + 9*y^3 + 9*y)*x^9/9 + (-y^10 + 10*y^7 - 15*y^4 - 10*y^2)*x^10/10 +...
where
exp(A(x,y)) = 1 + y*x + y*x^4 + y*x^9 + y*x^16 + y*x^25 +...
Triangle begins:
1;
0, -1;
0, 0, 1;
4, 0, 0, -1;
0, -5, 0, 0, 1;
0, 0, 6, 0, 0, -1;
0, 0, 0, -7, 0, 0, 1;
0, -4, 0, 0, 8, 0, 0, -1;
9, 0, 9, 0, 0, -9, 0, 0, 1;
0, -10, 0, -15, 0, 0, 10, 0, 0, -1;
0, 0, 11, 0, 22, 0, 0, -11, 0, 0, 1;
0, 0, 4, -12, 0, -30, 0, 0, 12, 0, 0, -1;
0, -13, 0, -13, 13, 0, 39, 0, 0, -13, 0, 0, 1;
0, 0, 28, 0, 28, -14, 0, -49, 0, 0, 14, 0, 0, -1;
0, 0, 0, -45, 0, -50, 15, 0, 60, 0, 0, -15, 0, 0, 1;
16, 0, 0, -4, 64, 0, 80, -16, 0, -72, 0, 0, 16, 0, 0, -1;
0, -17, 17, 0, 17, -85, 0, -119, 17, 0, 85, 0, 0, -17, 0, 0, 1;
0, -9, 18, -54, 0, -45, 108, 0, 168, -18, 0, -99, 0, 0, 18, 0, 0, -1;
0, 0, 19, -19, 114, 0, 95, -133, 0, -228, 19, 0, 114, 0, 0, -19, 0, 0, 1;
0, -20, 0, -30, 24, -200, 0, -175, 160, 0, 300, -20, 0, -130, 0, 0, 20, 0, 0, -1;
0, 0, 42, -21, 42, -42, 315, 0, 294, -189, 0, -385, 21, 0, 147, 0, 0, -21, 0, 0, 1;
0, 0, 22, -66, 88, -55, 88, -462, 0, -462, 220, 0, 484, -22, 0, -165, 0, 0, 22, 0, 0, -1;
0, 0, 0, -69, 92, -230, 69, -184, 644, 0, 690, -253, 0, -598, 23, 0, 184, 0, 0, -23, 0, 0, 1;
0, 0, 24, 0, 144, -124, 480, -84, 360, -864, 0, -990, 288, 0, 728, -24, 0, -204, 0, 0, 24, 0, 0, -1;
25, -25, 0, -75, 25, -250, 175, -875, 100, -655, 1125, 0, 1375, -325, 0, -875, 25, 0, 225, 0, 0, -25, 0, 0, 1; ...

Paul D. Hanna, Rows n=1..45, flattened as a list of n, a(n) for n = 1..1035.

Cf. A227311, A162552, A000203, A015128, A002318, A004404, A004405.

{T(n,k)=n*polcoeff(polcoeff(log(1+sum(m=1,sqrtint(n)+1,y*x^(m^2))+x*O(x^n)),n,x),k,y)}
for(n=1,25,for(k=1,n,print1(T(n,k),", "));print(""))

/* Alternate g.f., true for all m >= 0: */
{T(n,k,m=0) = if(k<1||m<0,0, (n/k/binomial(k+m,m)) * polcoeff(polcoeff( 1 - 1/(1+sum(j=1,sqrtint(n+1),y*x^(j^2))+x*O(x^n))^(m+1), n,x),k,y))}
for(n=1, 25, for(k=1, n, print1(T(n, k, 1), ", ")); print(""))

G.f.: Sum_{n>=1, k=1..n} T(n,k)*x^n*y^k*k*binomial(k+m,m)/n = 1 - 1/(1 + Sum_{n>=1} y*x^(n^2))^(m+1), which holds for all m >= 0.
Row sums equal A162552.
Sum_{k=1..n} T(n,k)*2^k = -(-1)^n*(sigma(2*n) - sigma(n)) for n>=1, where sigma is the sum of divisors of n, A000203.
Sum_{k=1..n} T(n,k)*2^k*k = -(-1)^n*n*A015128(n) for n>=1, where A015128(n) is the number of overpartitions of n, with g.f.: Product_{n>=1} (1+x^n)/(1-x^n).
Sum_{k=1..n} T(n,k)*2^k*k*(k+1) = -(-1)^n*4*n*A002318(n) for n>=1, where A002318 lists the coefficients in (1/theta_4(q)^2 -1)/4 in powers of q.
Sum_{k=1..n} T(n,k)*2^k*k*(k+1)*(k+2)/2! = -n*A004404(n) for n>=1, where A004404 lists the coefficients in 1/(1 + Sum_{n>=1} 2*x^(n^2))^3.
Sum_{k=1..n} T(n,k)*2^k*k*(k+1)*(k+2)*(k+3)/3! = -n*A004405(n) for n>=1, where A004405 lists the coefficients in 1/(1 + Sum_{n>=1} 2*x^(n^2))^4.
More generally:
Sum_{k=1..n} T(n,k)*y^k*k*binomial(k+m,m)/n = [x^n] 1 - 1/(1 + Sum_{n>=1} y*x^(n^2))^(m+1) for m>=0, n>=1.

A172259 Let CK(m) denote the complete elliptic integral of the first kind. a(n) is the n-th smallest integer k such that floor(CK(1/k)) = floor(CK(1/(k-1))) + 1.

Original entry on oeis.org

1, 2, 5, 14, 38, 101, 275, 746, 2026, 5507, 14969, 40689, 110604, 300652, 817255, 2221528, 6038739, 16414993, 44620576, 121291299, 329703934, 896228212, 2436200862, 6622280533, 18001224835, 48932402358, 133012060152, 361564266077, 982833574297, 2671618645410

Offset: 1

Michel Lagneau, Jan 30 2010

nonn

F(z,k) = Integral_{t=0..z} 1/(sqrt(1-t^2)*sqrt(1-k^2*t^2)) dt and the complete elliptic integral CK is defined by CK(k) = F(1,sqrt(1-k^2)). We calculate the values of CK(k) with k = 1/p, p = 1,2,3, ... and we propose a very interesting property: a(n+1)/a(n) tends toward e = 2.7182818... when n tends to infinity. For example, a(8) / a(7) = 2.718281581; a(9) / a(8) = 2.7182817562.

			a(3) = 38 because floor(CK(1/37)) = 4 and floor(CK(1/38)) = 5.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 575, Eq. 16.22.1 and 16.22.2.
M. Abramowitz and I. Stegun, "Elliptic Integrals", Chapter 17 of Handbook of Mathematical Functions. Dover Publications Inc., New York, 1046 p., (1965).
A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.

A. Cayley, An Elementary Treatise on Elliptic Functions, G. Bell and Sons, London, 1895, p. 56.
F. Clarke, The Taylor Series Coefficients of the Jacobi Elliptic Functions, slides. [broken link]

Cf. elliptic functions: A001936, A002318, A001937, A001934, A001938, A002754, A001939, A001940, A001941, A002753, A006089, A004005.

a0:=1:for p from 1 to 1000 do:a:= evalf(EllipticCK(1/p)):if floor(a)=a0+1 then print(p):a0:=floor(a):else fi:od:

F(z,k) = Integral_{t=0..z} 1/(sqrt(1-t^2)*sqrt(1-k^2*t^2)) dt. CK is defined by CK(k) = F(1,sqrt(1-k^2)). a(n) is the n-th integer k such that floor(CK(1/k)) = floor(CK(1/(k-1))) + 1.

cp's OEIS Frontend

A001934 Expansion of 1/theta_4(q)^2 in powers of q.

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A216273 Triangle generated by Sum_{n>=1, k=1..n} T(n,k)x^ny^k/n = log(1 + Sum_{n>=1} y*x^(n^2)), where coefficients are read by rows.

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A172259 Let CK(m) denote the complete elliptic integral of the first kind. a(n) is the n-th smallest integer k such that floor(CK(1/k)) = floor(CK(1/(k-1))) + 1.

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cp's OEIS Frontend

A001934 Expansion of 1/theta_4(q)^2 in powers of q.

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Julia

Maple

Mathematica

PARI

PARI

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Extensions

A216273 Triangle generated by Sum_{n>=1, k=1..n} T(n,k)*x^n*y^k/n = log(1 + Sum_{n>=1} y*x^(n^2)), where coefficients are read by rows.

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Author

Keywords

Examples

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PARI

PARI

Formula

A172259 Let CK(m) denote the complete elliptic integral of the first kind. a(n) is the n-th smallest integer k such that floor(CK(1/k)) = floor(CK(1/(k-1))) + 1.

Views

Author

Keywords

Comments

Examples

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Crossrefs

Programs

Maple

Formula

A216273 Triangle generated by Sum_{n>=1, k=1..n} T(n,k)x^ny^k/n = log(1 + Sum_{n>=1} y*x^(n^2)), where coefficients are read by rows.