A004404 -id:A004404 - 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.

A341364 Expansion of (1 / theta_4(x) - 1)^3 / 8.

Original entry on oeis.org

1, 6, 24, 77, 216, 552, 1315, 2964, 6387, 13255, 26640, 52074, 99336, 185430, 339483, 610709, 1081227, 1886484, 3247502, 5521365, 9279624, 15429149, 25397088, 41412030, 66928700, 107265576, 170556654, 269164346, 421765920, 656419080, 1015044526, 1559950185, 2383284894

Offset: 3

Ilya Gutkovskiy, Feb 10 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..10000

Cf. A002448, A004404, A014968, A015128, A063691, A319552, A327381, A338223, A341221, A341365, A341366, A341367, A341368, A341369, A341370.

g:= proc(n, i) option remember; `if`(n=0, 1/2, `if`(i=1, 0,
      g(n, i-1))+add(2*g(n-i*j, i-1), j=`if`(i=1, n, 1)..n/i))
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
      g(n$2)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 3):
seq(a(n), n=3..35);  # Alois P. Heinz, Feb 10 2021

nmax = 35; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^3/8, {x, 0, nmax}], x] // Drop[#, 3] &
nmax = 35; CoefficientList[Series[(1/8) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^3, {x, 0, nmax}], x] // Drop[#, 3] &

G.f.: (1/8) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^3.

a(n) ~ A319552(n)/8 ~ 3*exp(Pi*sqrt(3*n)) / (512*n^(3/2)). - Vaclav Kotesovec, Feb 20 2021

A288515 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^j)/(1 - x^j))^k.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 4, 0, 1, 6, 12, 8, 0, 1, 8, 24, 32, 14, 0, 1, 10, 40, 80, 76, 24, 0, 1, 12, 60, 160, 234, 168, 40, 0, 1, 14, 84, 280, 552, 624, 352, 64, 0, 1, 16, 112, 448, 1110, 1712, 1552, 704, 100, 0, 1, 18, 144, 672, 2004, 3912, 4896, 3648, 1356, 154, 0, 1, 20, 180, 960, 3346, 7896, 12600, 13120, 8184, 2532, 232, 0

Offset: 0

Ilya Gutkovskiy, Jun 10 2017

			Square array begins:
1,   1,    1,    1,     1,     1,  ...
0,   2,    4,    6,     8,    10,  ...
0,   4,   12,   24,    40,    60,  ...
0,   8,   32,   80,   160,   280,  ...
0,  14,   76,  234,   552,  1110,  ...
0,  24,  168,  624,  1712,  3913,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to partitions

Columns k=0-24 give: A000007, A015128, A001934, A004404 (alternating values), A284286, A004406-A004425 (alternating values).
Rows n=0-2 give: A000012, A005843, A046092.
Main diagonal gives A270919.
Antidiagonal sums give A299108.
Cf. A122141, A286815.

# JacobiTheta4 is defined in A002448.
A288515Column(k, len) = JacobiTheta4(len, -k)
for k in 0:8 A288515Column(k, 8) |> println end # Peter Luschny, Mar 12 2018

Table[Function[k, SeriesCoefficient[Product[((1 + x^i)/(1 - x^i))^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/EllipticTheta[4, 0, x]^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=1} ((1 + x^j)/(1 - x^j))^k.
G.f. of column k: 1/theta_4(x)^k, where theta_4() is the Jacobi theta function.
For asymptotics of column k see comment from Vaclav Kotesovec in A001934.

A319552 Expansion of 1/theta_4(q)^3 in powers of q = exp(Pi i t).

Original entry on oeis.org

1, 6, 24, 80, 234, 624, 1552, 3648, 8184, 17654, 36816, 74544, 147056, 283440, 535008, 990912, 1803882, 3232224, 5707624, 9943536, 17106960, 29088352, 48922320, 81438528, 134261584, 219336630, 355242288, 570675904, 909674688, 1439394192, 2261635168, 3529838208

Offset: 0

Seiichi Manyama, Sep 22 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

1/theta_4(q)^b: A015128 (b=1), A001934 (b=2), this sequence (b=3), A284286 (b=4), A319553 (b=8), A319554 (b=12).

Cf. A002131, A002448 (theta_4(q)), A004404, A213384.

N=99; x='x+O('x^N); Vec(prod(k=1, N, ((1-x^(2*k))/(1-x^k)^2)^3))

Convolution inverse of A213384.
a(n) = (-1)^n * A004404(n).
a(0) = 1, a(n) = (6/n)*Sum_{k=1..n} A002131(k)*a(n-k) for n > 0.
G.f.: Product_{k>=1} ((1 - x^(2k))/(1 - x^k)^2)^3.

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.

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

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A341364 Expansion of (1 / theta_4(x) - 1)^3 / 8.

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A288515 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^j)/(1 - x^j))^k.

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A319552 Expansion of 1/theta_4(q)^3 in powers of q = exp(Pi i t).

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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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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.