A015128 -id:A015128 - cp's OEIS frontend

A380582 a(n) = [x^n] G(x)^n, where G(x) = Product_{k >= 1} ((1 + x^k)/(1 - x^k))^(k^2) is the g.f. of A206622.

1, 2, 24, 236, 2432, 25752, 277152, 3019088, 33186816, 367378814, 4089875024, 45741207228, 513537853952, 5784253405192, 65332622356032, 739706089046736, 8392732289277952, 95401363286044260, 1086232605119042424, 12386037358495697292, 141422619808922418432, 1616691574828234720352

Offset: 0

Peter Bala, Jan 27 2025

Given an integer sequence {f(n) : n >= 0} with f(0) = 1, there is a unique power series F(x) with rational coefficients, where F(0) = 1, such that f(n) = [x^n] F(x)^n. F(x) is given by F(x) = series_reversion(x/E(x)), where E(x) = exp(Sum_{n >= 1} f(n)*x^n/n). Furthermore, if the series E(x) has integer coefficients then the series F(x) also has integer coefficients and the sequence {f(n)} satisfies the Gauss congruences: f(n*p^r) == f(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r (by Stanley, Ch. 5, Ex. 5.2(a), p. 72 and the Lagrange inversion formula).
Thus the present sequence satisfies the Gauss congruences. In fact, stronger congruences appear to hold for this sequence.
We conjecture that a(p) == 1 (mod p^3) for all primes p >= 5 (checked up to p = 61).
More generally, we conjecture that the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) holds for all primes p >= 5 and positive integers n and r. Some examples are given below.
Let A, B be integers and let C be a positive integer. Define u(n) = [x^(C*n)] Product_{k >= 1} ((1 + x^k)^A * (1 - x^k)^B)^(k^2). The present sequence is the case A = 1, B = -1 and C = 1. We conjecture that the above supercongruences also hold for the sequence {u(n)} for all primes p >= 7.

			Examples of supercongruences:
a(7) - a(1) = 3019088 - 2 = 2*(3^3)*(7^3)*163 == 0 (mod 7^3)
a(13) - a(1) = 5784253405192 - 2 = 2*5*(13^4)*20252279 == 0 (mod 13^4)
a(2*11) - a(2) = 18501616629347623668448 - 24 = (2^3)*(11^3)*17*1951*4243*9817*1257719 == 0 (mod 11^3)
a(5^2) - a(5) = 1884578634304981694792832319004 - 256504 = (2^2)*(5^6)*193381* 155926684363405438573 == 0 (mod 5^6)

Cf. A001158, A015128, A206622, A380290, A380581, A380583.

with(numtheory):
G(x) := series(exp(add( (1/4)*(sigma[3](2*k) - sigma[3](k))*x^k/k, k = 1..23 )),x,24):
seq(coeftayl(G(x)^n, x = 0, n), n = 0..23);

a(n) = [x^n] exp( n*Sum_{k >= 1} (sigma_3(2*k) - sigma_3(k))/4 * x^k/k ).

A004403 Expansion of 1/theta_3(q)^2 in powers of q.

Original entry on oeis.org

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

Euler transform of period 4 sequence [ -4,6,-4,2,...].

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.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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]

Cf. A001934, A015128, A096427.

# JacobiTheta3 is defined in A000122.
A004403List(len) = JacobiTheta3(len, -2)
A004403List(33) |> println # Peter Luschny, Mar 12 2018

CoefficientList[Series[1/EllipticTheta[3, 0, q]^2, {q, 0, 32}], q] (* Jean-François Alcover, Jul 18 2011 *)
QP = QPochhammer; s = QP[q^2]^2/QP[-q]^4 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)

{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 */

Expansion of (Sum x^(n^2), n = -inf .. inf )^(-2).
Expansion of elliptic function pi / 2K in powers of q.
G.f.: 1 / (Sum_{k} x^k^2)^2 = (Product_{k>0} (1 + x^(2k))^2 /((1-x^k)(1 + x^k)^3))^2.
a(n) = (-1)^n * A001934(n).
Conjecture: Sum_{k>=0} a(k) / exp(k*Pi) = (1/Pi^(1/2))*Gamma(3/4)^2 = A096427. - Simon Plouffe, Sep 09 2025

A096914 Number of partitions of 2*n into distinct parts with exactly two odd parts.

Original entry on oeis.org

1, 2, 4, 7, 11, 17, 25, 36, 50, 69, 93, 124, 163, 212, 273, 349, 442, 556, 695, 863, 1066, 1310, 1602, 1950, 2364, 2854, 3433, 4115, 4916, 5854, 6951, 8229, 9716, 11442, 13441, 15752, 18419, 21490, 25021, 29074, 33718, 39031, 45101, 52024, 59910

Offset: 2

Vladeta Jovovic, Aug 18 2004

Amrik Singh Nimbran and Paul Levrie, Series of the form Sum {a_n*binomial(2n, n)}, Math. Student (2023) Vol. 92, Nos. 3-4, 155-173. See p. 9.

Cf. A000009, A036469, A015128.

Drop[ Union[ CoefficientList[ Series[x^4* Product[1 + x^(2m), {m, 1, 50}] / Product[1 - x^(2m), {m, 1, 2}], {x, 0, 920}], x]], 1] (* Robert G. Wilson v, Aug 21 2004 *)
nmax = 50; Drop[CoefficientList[Series[(x^2/(1 - x - x^2 + x^3)) * Product[1 + x^m, {m, 1, nmax}], {x, 0, nmax}], x], 2] (* Vaclav Kotesovec, May 29 2018 *)

G.f. for number of partitions of n into distinct parts with exactly k odd parts is x^(k^2)*Product(1+x^(2*m), m=1..infinity)/Product(1-x^(2*m), m=1..k).

a(n) ~ 3^(3/4) * exp(Pi*sqrt(n/3)) * n^(1/4) / (2*Pi^2). - Vaclav Kotesovec, May 29 2018

More terms from Robert G. Wilson v, Aug 21 2004

A101277 Number of partitions of 2n in which all odd parts occur with multiplicity 2. There is no restriction on the even parts.

Original entry on oeis.org

1, 2, 3, 6, 10, 16, 25, 38, 57, 84, 121, 172, 243, 338, 465, 636, 862, 1158, 1546, 2050, 2702, 3542, 4616, 5986, 7729, 9932, 12707, 16196, 20563, 26010, 32788, 41194, 51591, 64418, 80195, 99558, 123269, 152226, 187514, 230434, 282519, 345596, 421844, 513834

Offset: 0

Noureddine Chair, Dec 20 2004; revised Jan 05 2005

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
This is also A080054 times 1/Product_{k>=1} (1 - x^(2k)).
There are no partitions of 2n+1 in which all odd parts occur with multiplicity 2. - Michael Somos, Oct 27 2008

			G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 10*x^4 + 16*x^5 + 25*x^6 + 38*x^7 + 57*x^8 + ...
G.f. = 1/q + 2*q^11 + 3*q^23 + 6*q^35 + 10*q^47 + 16*q^59 + 25*q^71 + ...
E.g. 12 = 10 + 2 = 10 + 1 + 1 = 8 + 4 = 8 + 2 + 2 = 8 + 2 + 1 + 1 = 6 + 6 = 6 + 4 + 2 = 6 + 4 + 1 + 1 = 6 + 3 + 3 = 6 + 2 + 2 + 2 = 6 + 2 + 2 + 1 + 1 = 5 + 5 + 2 = 5 + 5 + 1 + 1 = 4 + 4 + 4 = 4 + 4 + 2 + 2 = 4 + 4 + 2 + 1 + 1 = 4 + 3 + 3 + 2 = 4 + 3 + 3 + 1 + 1 = 4 + 2 + 2 + 2 + 2 = 4 + 2 + 2 + 2 + 1 + 1 = 3 + 3 + 2 + 2 + 2 = 3 + 3 + 2 + 2 + 1 + 1 = 2 + 2 + 2 + 2 + 2 + 2 = 2 + 2 + 2 + 2 + 2 + 1 + 1.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Cristina Ballantine, Mircea Merca, Jacobi's Four and Eight Squares Theorems and Partitions into Distinct Parts, Mediterranean Journal of Mathematics (2019) Vol. 16, No. 2, 26.
Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol, Ramanujan Theta Functions

Cf. A015128, A098151, A080054.

series(product(1/((1-x^(2*k-1))^2*(1-x^(4*k))),k=1..100),x=0,100);

nmax=50; CoefficientList[Series[Product[1/((1-x^(2*k-1))^2 * (1-x^(4*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 30 2015 *)
(2/(QPochhammer[x] QPochhammer[-1, -x]) + O[x]^45)[[3]] (* Vladimir Reshetnikov, Nov 22 2016 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^2] / QPochhammer[ x]^2, {x, 0, n}]; (* Michael Somos, Nov 22 2016 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Nov 22 2016 *)
a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x, -x] QPochhammer[ x]), {x, 0, n}]; (* Michael Somos, Nov 22 2016 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)), n))}; /* Michael Somos, Feb 10 2005 */

Euler transform of period 4 sequence [2, 0, 2, 1, ...]. - Michael Somos, Feb 10 2005
G.f.: (1/theta_4(0, x))*Product_{k>0}(1+x^(2k)) = theta_4(0, x^2)/theta_4(0, x)*Product_{k>0}(1-x^(2k)) = 1/Product_{k>0} ((1-x^(2k-1))^2 * (1-x^(4k))).
Expansion of 1 / (psi(-x) * chi(-x)) in powers of x where psi(), chi() are Ramanujan theta functions. - Michael Somos, Oct 27 2008
Expansion of q^(1/12) * eta(q^2)^2 / (eta(q)^2 * eta(q^4)) in powers of q. - Michael Somos, Oct 27 2008
a(n) ~ sqrt(5) * exp(Pi*sqrt(5*n/6)) / (8*sqrt(3)*n). - Vaclav Kotesovec, Aug 30 2015
G.f.: 2/((x; x)inf * (-1; -x)_inf), where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 22 2016
Expansion of phi(-x^2) / f(-x)^2 = chi(x) / f(-x) = 1 / (chi(-x)^2 * f(-x^4)) = f(-x^4) / psi(-x)^2 = psi(-x) / chi(-x) = chi(x)^2 / psi(-x^2) in powers of x. - Michael Somos, Nov 22 2016

A195585 sigma(2*n^2) - sigma(n^2).

Original entry on oeis.org

2, 8, 26, 32, 62, 104, 114, 128, 242, 248, 266, 416, 366, 456, 806, 512, 614, 968, 762, 992, 1482, 1064, 1106, 1664, 1562, 1464, 2186, 1824, 1742, 3224, 1986, 2048, 3458, 2456, 3534, 3872, 2814, 3048, 4758, 3968, 3446, 5928, 3786, 4256, 7502, 4424, 4514, 6656, 5602, 6248, 7982

Offset: 1

Paul D. Hanna, Sep 20 2011

nonn

			L.g.f.: L(x) = 2*x + 8*x^2/2 + 26*x^3/3 + 32*x^4/4 + 62*x^5/5 + 104*x^6/6 +...
where the g.f. of A195584 begins:
exp(L(x)) = 1 + 2*x + 6*x^2 + 18*x^3 + 42*x^4 + 102*x^5 + 238*x^6 +...

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Cf. A195584, A215603, A054785, A015128, A002117.

Table[DivisorSigma[1,2n^2]-DivisorSigma[1,n^2],{n,60}] (* Harvey P. Dale, May 05 2021 *)

PARI
```
{a(n)=sigma(2*n^2)-sigma(n^2)}
```

Equals the logarithmic derivative of A195584.
a(n) = A054785(n^2), where A054785 is the logarithmic derivative of A015128, which is the number of overpartitions of n.
Sum_{k=1..n} a(k) ~ c * n^3, where c = 7*zeta(3)/Pi^2 = 0.85255679763501158184... . - Amiram Eldar, Mar 17 2024

A201079 Irregular triangle read by rows: number of {0,2,4,6...}-shifted Schroeder paths of length n and area k.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 2, 3, 3, 0, 1, 2, 4, 6, 7, 7, 5, 0, 0, 1, 2, 4, 7, 11, 14, 18, 20, 19, 15, 8, 0, 0, 1, 2, 4, 8, 12, 19, 26, 35, 43, 52, 57, 61, 57, 46, 30, 13, 0, 0, 0, 1, 2, 4, 8, 13, 21, 32, 45, 61, 81, 101, 125, 146, 167, 183, 194, 191, 178, 146, 103, 58, 21, 0, 0, 0

Offset: 0

N. J. A. Sloane, Nov 26 2011

			Triangle begins
1
1
1 2 0
1 2 3 3 0
1 2 4 6 7 7 5 0 0
1 2 4 7 11 14 18 20 19 15 8 0 0
1 2 4 8 12 19 26 35 43 52 57 61 57 46 30 13 0 0 0
...

Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.

Row sums give A063020. Rows converge to A015128.

Cf. S-shifted Schroeder paths for various S: A201075 {0,1}, A201076 {0,2}, A201080 {0,1,3,5...}, A201159 {0,1,2}.

max = 8; s0 = Range[2, max, 2];
gf = Expand /@ FixedPoint[With[{g = Normal@#}, 1 + q x g (g /. {x :> q^2 x}) + Sum[q^(j^2 - j) x^j Product[g /. {x :> q^(2 i - 2) x}, {i, j}], {j, s0}] + O[x]^max] &, 0];
Flatten[Reverse[CoefficientList[#, q]][[;; ;; 2]] & /@ CoefficientList[gf, x]] (* Andrey Zabolotskiy, Jan 02 2024 *)

Name and rows 3 and 5 corrected and row 7 added by Andrey Zabolotskiy, Jan 02 2024

A233758 Bisection of A006950 (the even part).

Original entry on oeis.org

1, 1, 3, 5, 10, 16, 28, 43, 70, 105, 161, 236, 350, 501, 722, 1016, 1431, 1981, 2741, 3740, 5096, 6868, 9233, 12306, 16357, 21581, 28394, 37128, 48406, 62777, 81182, 104494, 134131, 171467, 218607, 277691, 351841, 444314, 559727, 703002, 880896, 1100775

Offset: 1

Omar E. Pol, Jan 11 2014

nonn

See Zaletel-Mong paper, page 14, FIG. 11: C2a is this sequence, C2b is A233759, C2c is A015128.

M. P. Zaletel and R. S. K. Mong, Exact Matrix Product States for Quantum Hall Wave Functions, arXiv preprint arXiv:1208.4862 [cond-mat.str-el] (2012), 14 (C2a).

Cf. A000009, A015128, A006950, A069910, A069911, A195825, A195836, A211970, A233759.

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, b[n - i, i - Mod[i, 2]]]]];
a[n_] := b[2 n - 2, 2 n - 2];
Table[a[n], {n, 1, 42}] (* Jean-François Alcover, Dec 11 2018, after Alois P. Heinz in A006950 *)

A233759 Bisection of A006950 (the odd part).

Original entry on oeis.org

1, 2, 4, 7, 13, 21, 35, 55, 86, 130, 196, 287, 420, 602, 858, 1206, 1687, 2331, 3206, 4368, 5922, 7967, 10670, 14193, 18803, 24766, 32490, 42411, 55159, 71416, 92152, 118434, 151725, 193676, 246491, 312677, 395537, 498852, 627509, 787171, 985043, 1229494

Offset: 1

Omar E. Pol, Jan 11 2014

nonn

See Zaletel-Mong paper, page 14, FIG. 11: C2a is A233758, C2b is this sequence, C2c is A015128.

M. P. Zaletel and R. S. K. Mong, Exact Matrix Product States for Quantum Hall Wave Functions, arXiv preprint arXiv:1208.4862 [cond-mat.str-el] (2012), 14 (C2b).

Cf. A000009, A015128, A006950, A069910, A069911, A195825, A195836, A211970, A233758.

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, b[n - i, i - Mod[i, 2]]]]];
a[n_] := b[2 n - 1, 2 n - 1];
Table[a[n], {n, 1, 42}] (* Jean-François Alcover, Dec 11 2018, after Alois P. Heinz in A006950 *)

A237044 Number of overcompositions of n minus the number of partitions of n.

Original entry on oeis.org

0, 1, 2, 9, 21, 53, 133, 309, 706, 1572, 3534, 7752, 16991, 36807, 79385, 170528, 364563, 776739, 1649071, 3490698, 7366917, 15512544, 32583646, 68306009, 142902505, 298446956, 622232624, 1295316994, 2692580198, 5589582431, 11588900240, 23999045850

Offset: 0

Omar E. Pol, Feb 02 2014

nonn

Cf. A000041, A011782, A015128, A056823, A230441, A236002, A236633, A237045, A235047, A237272.

a(n) = A236002(n) - A000041(n).

A237045 Number of overcompositions of n minus the number of overpartitions of n.

Original entry on oeis.org

0, 0, 0, 4, 12, 36, 104, 260, 628, 1448, 3344, 7464, 16564, 36180, 78480, 169232, 362732, 774172, 1645508, 3485788, 7360208, 15503432, 32571360, 68289536, 142880552, 298417848, 622194236, 1295266596, 2692514348, 5589496748, 11588789220, 23998902548

Offset: 0

Omar E. Pol, Feb 02 2014

nonn

Number of overcompositions of n that contain at least two parts in increasing order.

			Illustration of a(4) = -6 with both overcompositions and overpartitions in colexicographic order.
--------------------------------------------------------
.    Overcompositions of 4      Overpartitions of 4
--------------------------------------------------------
.    _ _ _ _                    _ _ _ _
1   |.| | | |  1', 1,  1,  1   |.| | | |  1', 1,  1,  1
2   |_| | | |  1,  1,  1,  1   |_| | | |  1,  1,  1,  1
3   |  .|.| |  2', 1', 1       |  .|.| |  2', 1', 1
4   |   |.| |  2,  1', 1       |   |.| |  2,  1', 1
5   |  .| | |  2', 1,  1       |  .| | |  2', 1,  1
6   |_ _| | |  2,  1,  1       |_ _| | |  2,  1,  1
7  *|.|  .| |  1', 2', 1       |    .|.|  3', 1
8  *| |  .| |  1,  2', 1       |     |.|  3,  1
9  *|.|   | |  1', 2,  1       |    .| |  3', 1
10 *|_|   | |  1,  2,  1       |_ _ _| |  3,  1
11  |    .|.|  3', 1'          |  .|   |  2', 2
12  |     |.|  3,  1'          |_ _|   |  2,  2
13  |    .| |  3', 1           |      .|  4'
14  |_ _ _| |  3,  1           |_ _ _ _|  4
15 *|.| |  .|  1', 1,  2'
16 *| | |  .|  1,  1,  2'
17 *|.| |   |  1', 1,  2
18 *|_| |   |  1,  1,  2
19  |  .|   |  2', 2
20  |_ _|   |  2,  2
21 *|.|    .|  1', 3'
22 *| |    .|  1,  3'
23 *|.|     |  1', 3
24 *|_|     |  1,  3
25  |      .|  4'
26  |_ _ _ _|  4
.
There are 26 overcompositions of 4 and there are 14 overpartitions of 4, so the difference is a(4) = 26 - 14 = 12.
On the other hand there are 12 overcompositions of 4 that contain at least two parts in increasing order, so a(4) = 12.

Cf. A000041, A011782, A015128, A056823, A230441, A236002, A236633, A237044, A237047, A237272.

a(n) = A236002(n) - A015128(n).

cp's OEIS Frontend

A380582 a(n) = [x^n] G(x)^n, where G(x) = Product_{k >= 1} ((1 + x^k)/(1 - x^k))^(k^2) is the g.f. of A206622.

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A004403 Expansion of 1/theta_3(q)^2 in powers of q.

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A096914 Number of partitions of 2*n into distinct parts with exactly two odd parts.

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Extensions

A101277 Number of partitions of 2n in which all odd parts occur with multiplicity 2. There is no restriction on the even parts.

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Maple

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A195585 sigma(2*n^2) - sigma(n^2).

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A201079 Irregular triangle read by rows: number of {0,2,4,6...}-shifted Schroeder paths of length n and area k.

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Mathematica

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A233758 Bisection of A006950 (the even part).

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A233759 Bisection of A006950 (the odd part).

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