A010816 -id:A010816 - cp's OEIS frontend

A369711 Maximum coefficient of (1 - x)^3 * (1 - x^2)^3 * (1 - x^3)^3 * ... * (1 - x^n)^3.

1, 3, 8, 15, 44, 50, 117, 186, 356, 561, 969, 1761, 3508, 5789, 10347, 19023, 35580, 62388, 111255, 205653, 376496, 674085, 1201809, 2211462, 4056220, 7287672, 13027698, 24005627, 43800562, 79033269, 141583272, 260061408, 473603594, 855436899, 1532383878, 2813222766

Offset: 0

Ilya Gutkovskiy, Jan 29 2024

nonn

Cf. A010816, A086376, A369709.

Table[Max[CoefficientList[Product[(1 - x^k)^3, {k, 1, n}], x]], {n, 0, 35}]

a(n) = vecmax(Vec(prod(i=1, n, (1-x^i)^3))); \\ Michel Marcus, Jan 29 2024

from collections import Counter
def A369711(n):
    c = {0:1}
    for k in range(1,n+1):
        d = Counter(c)
        for j in c:
            a = c[j]
            d[j+k] -= 3*a
            d[j+2*k] += 3*a
            d[j+3*k] -= a
        c = d
    return max(c.values()) # Chai Wah Wu, Feb 07 2024

A171805 G.f.: Series reversion of x/P(x)^3 where P(x) is the g.f. for Partition numbers (A000041).

Original entry on oeis.org

1, 3, 18, 130, 1044, 8946, 80135, 741312, 7027515, 67911855, 666525630, 6625647054, 66570488901, 674964968175, 6897258376218, 70961851119848, 734455079297433, 7641851681095236, 79886815507105175, 838655487787502616, 8837797224686207976, 93454820274339167191

Offset: 1

Paul D. Hanna, Dec 20 2009

nonn

			G.f.: A(x) = x + 3*x^2 + 18*x^3 + 130*x^4 + 1044*x^5 + 8946*x^6 +...
where Series_Reversion(A(x)) = x/P(x)^3 = x*eta(x)^3 and
x*eta(x)^3 = x - 3*x^2 + 5*x^4 - 7*x^7 + 9*x^11 - 11*x^16 + 13*x^22 +...

Cf. A109085, A171802, A171803, A171804, A000041, A007312, A010815, A010816.

InverseSeries[x QPochhammer[x]^3 + O[x]^30][[3]] (* Vladimir Reshetnikov, Nov 21 2016 *)
(* Calculation of constants {d,c}: *) eq = FindRoot[{r/QPochhammer[s]^3 == s, 1/s + 3*(s/r)^(1/3)*Derivative[0, 1][QPochhammer][s, s] == (3*(Log[1 - s] + QPolyGamma[0, 1, s]))/(s*Log[s])}, {r, 1/10}, {s, 1/8}, WorkingPrecision -> 1000]; {N[1/r /. eq, 120], val = Sqrt[r*(-1 + s)*s^2*(Log[s]^2/(6*Pi*(r*(-4*s*ArcTanh[1 - 2*s] + Log[1 - s]*(2 + 3*(-1 + s)*Log[1 - s] + Log[s] - s*Log[s])) - (-1 + s)*(-3*r*QPolyGamma[0, 1, s]^2 + r*QPolyGamma[1, 1, s] + QPolyGamma[0, 1, s]*(r*(2 - 6*Log[1 - s] + Log[s]) + 6*(r/s)^(2/3)*s^2*Log[s]* Derivative[0, 1][QPochhammer][s, s]) + s*Log[s]*((r/s)^(1/3)*s*(6*(r/s)^(1/3) * Log[1 - s] * Derivative[0, 1][QPochhammer][s, s] - 4*s*Log[s] * Derivative[0, 1][QPochhammer][s, s]^2 + (r/s)^(1/3)*s*Log[s]* Derivative[0, 2][QPochhammer][s, s]) - 2*r*Derivative[0, 0, 1][ QPolyGamma][0, 1, s])))))] /. eq; N[Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3]} (* Vaclav Kotesovec, Oct 03 2023 *)

{a(n)=polcoeff(serreverse(x*eta(x+x*O(x^n))^3),n)}

G.f. A(x) satisfies:
(1) A(x) = x/Product_{n>=1} (1 - A(x)^n)^3 ;
(2) A(x) = x/Sum_{n>=0} (-1)^n*(2n+1)*A(x)^(n(n+1)/2).
G.f.: A(x) = Series_Reversion(x*eta(x)^3) where eta(q) is the q-expansion of the Dedekind eta function without the q^(1/24) factor (A010815).
Self-convolution cube of A171804 (with offset).
a(n) ~ c * d^n / n^(3/2), where d = 11.34340769381039824727582112969136186... and c = 0.05972244738388663765328174469956... - Vaclav Kotesovec, Nov 11 2017

More terms from Vladimir Reshetnikov, Nov 21 2016

A198954 Expansion of the rotational partition function for a heteronuclear diatomic molecule.

Original entry on oeis.org

1, 3, 0, 5, 0, 0, 7, 0, 0, 0, 9, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0, 19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 27, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Michael Somos, Oct 31 2011

The partition function of a heteronuclear diatomic molecule is Sum_{J>=0} (2*J + 1) * exp( - J * (J + 1) * hbar^2 / (2 * I * k * T)) where I is the moment of inertia, hbar is reduced Planck's constant, k is Boltzmann's constant, and T is temperature. The degeneracy for the J-th energy level is 2*J + 1.
As triangle: triangle T(n,k), read by rows, given by (3,-4/3,1/3,0,0,0,0,0,0,0,...) DELTA (0,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 01 2011
Note that the g.f. theta_1'(0, q^(1/2)) / (2 * q^(1/8)) = 1 - 3*q  + 5*q^3 - 7*q^6 + 9*q^10 + ... which is the same as this sequence except the signs alternate. - Michael Somos, Aug 26 2015

			G.f. = 1 + 3*x + 5*x^3 + 7*x^6 + 9*x^10 + 11*x^15 + 13*x^21 + 15*x^28 + ...
G.f. = 1 + 3*q^2 + 5*q^6 + 7*q^12 + 9*q^20 + 11*q^30 + 13*q^42 + 15*q^56 + ...
Triangle begins:
   1;
   3, 0;
   5, 0, 0;
   7, 0, 0, 0;
   9, 0, 0, 0, 0;
  11, 0, 0, 0, 0, 0;
  13, 0, 0, 0, 0, 0, 0;
  15, 0, 0, 0, 0, 0, 0, 0;
  17, 0, 0, 0, 0, 0, 0, 0, 0;

G. H. Wannier, Statistical Physics, Dover Publications, 1987, see p. 215 equ. (11.13).

Antti Karttunen, Table of n, a(n) for n = 0..10011

Cf. A053187, A107270.

seq(op([2*i+1,0$i]), i=0..10); # Robert Israel, Jan 15 2015

a[ n_] := If[ n < 0, 0, With[ {m = Sqrt[8 n + 1]}, If[ IntegerQ[m], m KroneckerSymbol[ 4, m], 0]]]; (* Michael Somos, Aug 26 2015 *)

{a(n) = my(m); if( issquare( 8*n + 1, &m), m, 0)};

G.f.: Sum_{k>=0} (2*k + 1) * x^( (k^2 + k) / 2). This is related to Jacobi theta functions.
a(n) = (t*(t+1)-2*n-1)*(t-r), where t = floor(sqrt(2*(n+1))+1/2) and r = floor(sqrt(2*n)+1/2). - Mikael Aaltonen, Jan 15 2015
a(n) = A053187(2n+1) - A053187(2n). - Robert Israel, Jan 15 2015
a(n) = abs(A010816(n)). - Joerg Arndt, Jan 16 2015

A319933 A(n, k) = [x^k] DedekindEta(x)^n, square array read by descending antidiagonals, A(n, k) for n >= 0 and k >= 0.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, -1, -2, 1, 0, 0, -1, -3, 1, 0, 0, 2, 0, -4, 1, 0, 1, 1, 5, 2, -5, 1, 0, 0, 2, 0, 8, 5, -6, 1, 0, 1, -2, 0, -5, 10, 9, -7, 1, 0, 0, 0, -7, -4, -15, 10, 14, -8, 1, 0, 0, -2, 0, -10, -6, -30, 7, 20, -9, 1, 0, 0, -2, 0, 8, -5, 0, -49, 0, 27, -10, 1

Offset: 0

Peter Luschny, Oct 02 2018

The columns are generated by polynomials whose coefficients constitute the triangle of signed D'Arcais numbers A078521 when multiplied with n!.

			[ 0] 1,   0,   0,    0,     0,    0,     0,     0,     0,     0, ... A000007
[ 1] 1,  -1,  -1,    0,     0,    1,     0,     1,     0,     0, ... A010815
[ 2] 1,  -2,  -1,    2,     1,    2,    -2,     0,    -2,    -2, ... A002107
[ 3] 1,  -3,   0,    5,     0,    0,    -7,     0,     0,     0, ... A010816
[ 4] 1,  -4,   2,    8,    -5,   -4,   -10,     8,     9,     0, ... A000727
[ 5] 1,  -5,   5,   10,   -15,   -6,    -5,    25,    15,   -20, ... A000728
[ 6] 1,  -6,   9,   10,   -30,    0,    11,    42,     0,   -70, ... A000729
[ 7] 1,  -7,  14,    7,   -49,   21,    35,    41,   -49,  -133, ... A000730
[ 8] 1,  -8,  20,    0,   -70,   64,    56,     0,  -125,  -160, ... A000731
[ 9] 1,  -9,  27,  -12,   -90,  135,    54,   -99,  -189,   -85, ... A010817
[10] 1, -10,  35,  -30,  -105,  238,     0,  -260,  -165,   140, ... A010818
    A001489,  v , A167541, v , A319931,  v ,         diagonal: A008705
           A080956       A319930      A319932

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. Fifth ed., Clarendon Press, Oxford, 2003.

M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
Steven R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
Vaclav Kotesovec, The integration of q-series
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
M. Newman, A table of the coefficients of the powers of eta(tau), Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216. [Annotated scanned copy]
Tim Silverman, Counting Cliques in Finite Distant Graphs, arXiv preprint arXiv:1612.08085 [math.CO], 2016.
Michael Somos, Introduction to Ramanujan theta functions
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Transpose of A286354.

Cf. A078521, A319574 (JacobiTheta3).

# DedekindEta is defined in A000594
for n in 0:10
    DedekindEta(10, n) |> println
end

DedekindEta := (x, n) -> mul(1-x^j, j=1..n):
A319933row := proc(n, len) series(DedekindEta(x, len)^n, x, len+1):
seq(coeff(%, x, j), j=0..len-1) end:
seq(print([n], A319933row(n, 10)), n=0..10);

eta[x_, n_] := Product[1 - x^j, {j, 1, n}];
A[n_, k_] := SeriesCoefficient[eta[x, k]^n, {x, 0, k}];
Table[A[n - k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 10 2018 *)

from sage.modular.etaproducts import qexp_eta
def A319933row(n, len):
    return (qexp_eta(ZZ['q'], len+4)^n).list()[:len]
for n in (0..10):
    print(A319933row(n, 10))

A202394 Expansion of f(-x)^3 + 9 * x * f(-x^9)^3 in powers of x where f() is a Ramanujan theta function.

Original entry on oeis.org

1, 6, 0, 5, 0, 0, -7, 0, 0, 0, -18, 0, 0, 0, 0, -11, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0, -19, 0, 0, 0, 0, 0, 0, 0, 0, 0, -42, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 54, 0, 0, 0, 0

Offset: 0

Michael Somos, Dec 18 2011

sign

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 + 6*x + 5*x^3 - 7*x^6 - 18*x^10 - 11*x^15 + 13*x^21 + 30*x^28 + ...
G.f. = q + 6*q^9 + 5*q^25 - 7*q^49 - 18*q^81 - 11*q^121 + 13*q^169 + 30*q^225 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A010816, A116916.

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^3 + 9 x QPochhammer[ x^9]^3, {x, 0, n}]; (* Michael Somos, May 26 2014 *)

{a(n) = local(m); if( issquare(8*n + 1, &m), (-1)^(m \ 6) * m * ((m%3 == 0) + 1), 0)};

{a(n) = local(A); if( n<0, 0,  A = x * O(x^n); polcoeff( eta(x + A)^3 + 9 * x * eta(x^9 + A)^3, n))};

Expansion of f(-x^3) * a(x) in powers of x where f() is a Ramanujan theta function and a() is a cubic AGM theta function.
Expansion of q^(-1/8) * (eta(q)^3 + 9 * eta(q^9)^3) in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 41472^(1/2) (t / i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A116916.
G.f.: Sum_{k} -(-1)^k * (6*k - 1) * x^(3*k*(3*k - 1)/2) + Sum_{k>0} -(-1)^k * 6 * (2*k - 1) * x^(9*k*(k - 1)/2 + 1).
a(3*n + 2) = a(5*n + 2) = a(5*n + 4) = a(9*n + 4) = a(9*n + 7) = 0. a(3*n) = A116916(n). a(9*n + 1) = 6 * A010816(n). a(25*n + 3) = 5 * a(n).
a(n) nonzero if and only if n is a triangular number.

A215597 Expansion of psi(-x) * f(-x)^3 in powers of x where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, -4, 3, 4, -2, 0, -11, 4, 0, 12, 10, -12, -7, -4, 0, -12, 16, 0, 6, 0, 9, 8, -10, 0, -18, -20, 0, 20, -14, 12, 11, 24, 0, 0, -22, 0, 16, -20, -6, -12, 0, 0, -3, 4, 0, -20, 48, 0, 14, 28, 0, -40, 0, 0, 0, -8, -33, -4, -26, 0, 30, 28, 0, 0, 2, 12, -16, 20, 0

Offset: 0

Michael Somos, Aug 16 2012

sign

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

			1 - 4*x + 3*x^2 + 4*x^3 - 2*x^4 - 11*x^6 + 4*x^7 + 12*x^9 + 10*x^10 + ...
q - 4*q^5 + 3*q^9 + 4*q^13 - 2*q^17 - 11*q^25 + 4*q^29 + 12*q^37 + 10*q^41 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A010816, A106459, A215596.

A215597[n_] := SeriesCoefficient[(QPochhammer[x]^4 * QPochhammer[x^4])/ QPochhammer[x^2], {x, 0, n}]; Table[A215597[n], {n, 0, 50}] (* G. C. Greubel, Oct 01 2017 *)

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

Expansion of q^(-1/4) * eta(q)^4 * eta(q^4) / eta(q^2) in powers of q.
Euler transform of period 4 sequence [ -4, -3, -4, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (128 t)) = 2^(19/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A215596.
a(n) = (-1)^floor( n/2 ) * b(4*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * (-p)^(e/2) if p == 3 (mod 4),
Convolution of A106459 and A010816.

A227317 Expansion of psi(x)^6 * phi(-x)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, -5, -10, 5, 6, 10, 40, -20, -50, 19, -52, -30, 50, -25, 74, 97, 50, -25, -140, 69, -34, -100, -50, -185, -6, 83, 310, -60, -60, 410, -128, 145, -100, -245, 250, -87, -90, -400, -410, -151, 362, 185, -50, 285, 30, 150, -240, 500, 370, -68, 222, 5, -190

Offset: 0

Michael Somos, Sep 02 2013

sign

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

			1 + 2*x - 5*x^2 - 10*x^3 + 5*x^4 + 6*x^5 + 10*x^6 + 40*x^7 - 20*x^8 + ...
q^3 + 2*q^7 - 5*q^11 - 10*q^15 + 5*q^19 + 6*q^23 + 10*q^27 + 40*q^31 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A008439, A010816, A215600, A227695.

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2]^5 / QPochhammer[ q])^2, {q, 0, n}]

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 / eta(x + A))^2, n))}

Expansion of psi(x)^5 * f(-x)^3 = psi(x)^2 * f(-x^2)^6 in powers of x where psi(), f() are Ramanujan theta functions.
Expansion of q^(-3/4) * (eta(q^2)^5 / eta(x))^2 in powers of q.
Euler transform of period 2 sequence [ 2, -8, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 128 (t / i)^4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A227695.
G.f.: (Product_{k>0} (1 - x^(2*k))^5 / (1 - x^k))^2.
Convolution of A008439 and A010816.
-8 * a(n) = A215600(2*n + 1).

A369790 Number of different coefficient values in expansion of Product_{k=1..n} (1-x^k)^3.

Original entry on oeis.org

1, 4, 5, 15, 13, 31, 26, 57, 42, 91, 66, 139, 95, 209, 129, 283, 171, 365, 216, 463, 272, 573, 333, 697, 401, 825, 468, 993, 545, 1139, 629, 1315, 725, 1509, 815, 1689, 920, 1921, 1030, 2139, 1147, 2367, 1261, 2619, 1391, 2861, 1521, 3135, 1659, 3409, 1802, 3703, 1952

Offset: 0

Seiichi Manyama, Feb 01 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A010816, A039822.

a(n) = #Set(Vec(prod(k=1, n, (1-x^k)^3)));

from collections import Counter
def A369790(n):
    c = {0:1}
    for k in range(1,n+1):
        d = Counter(c)
        for j in c:
            a = c[j]
            d[j+k] -= 3*a
            d[j+2*k] += 3*a
            d[j+3*k] -= a
        c = d
    return len(set(c.values()))+int(max(c)+1>len(c)) # Chai Wah Wu, Feb 01 2024

A274094 a(0)=0; thereafter (-1)^(n+1)*n appears n times.

Original entry on oeis.org

0, 1, -2, -2, 3, 3, 3, -4, -4, -4, -4, 5, 5, 5, 5, 5, -6, -6, -6, -6, -6, -6, 7, 7, 7, 7, 7, 7, 7, -8, -8, -8, -8, -8, -8, -8, -8, 9, 9, 9, 9, 9, 9, 9, 9, 9, -10, -10, -10, -10, -10, -10, -10, -10, -10, -10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, -12, -12, -12, -12, -12, -12, -12, -12, -12, -12

Offset: 0

N. J. A. Sloane, Jun 10 2016

sign

Chai Wah Wu, Table of n, a(n) for n = 0..10011

Cf. A002024, A274093.

Partial sums of A010816.

s1:=[0];
for n from 1 to 15 do
for i from 1 to n do
s1:=[op(s1),(-1)^(n+1)*n]; od: od:
s1;

Join[{0}, Flatten[Table[(-1)^(n+1)*n, {n, 15}, {n}]]] (* Paolo Xausa, Sep 30 2024 *)
Join[{0},Flatten[Table[PadRight[{},n,(-1)^(n+1) n],{n,15}]]] (* Harvey P. Dale, Nov 28 2024 *)

A274094_list = [0]+ [i for n in range(1,142) for i in [n if n % 2 else -n]*n] # Chai Wah Wu, Jun 11 2016

from math import isqrt
def A274094(n): return m if (m:=isqrt(n<<3)+1>>1)&1 else -m # Chai Wah Wu, Nov 05 2024

A303992 Triangular array T(n,k) giving coefficients in expansion of Product_{j=1..n} (1-x^j)^3.

Original entry on oeis.org

1, 1, -3, 3, -1, 1, -3, 0, 8, -6, -6, 8, 0, -3, 1, 1, -3, 0, 5, 3, -6, -13, 9, 15, 0, -15, -9, 13, 6, -3, -5, 0, 3, -1, 1, -3, 0, 5, 0, 3, -13, -6, 9, 9, 24, -21, -24, -9, 3, 44, 3, -9, -24, -21, 24, 9, 9, -6, -13, 3, 0, 5, 0, -3, 1

Offset: 0

Seiichi Manyama, May 04 2018

			Irregular triangle starts:
n\k| 0   1  2   3   4   5    6  7   8  9   10  11  12 13  14  15 16 17  18
---+-----------------------------------------------------------------------
0  | 1;
1  | 1, -3, 3, -1;
2  | 1, -3, 0,  8, -6, -6,   8, 0, -3, 1;
3  | 1, -3, 0,  5,  3, -6, -13, 9, 15, 0, -15, -9, 13, 6, -3, -5, 0, 3, -1;

Seiichi Manyama, Rows n = 0..26, flattened

Cf. A010816, A231599, A304080.

T(n, k) = polcoef(prod(j=1, n, (1-x^j)^3), k);
tabf(nn) = for(n=0, nn, for(k=0, 3*n*(n+1)/2, print1(T(n, k), ", ")); print)

cp's OEIS Frontend

A369711 Maximum coefficient of (1 - x)^3 * (1 - x^2)^3 * (1 - x^3)^3 * ... * (1 - x^n)^3.

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Python

A171805 G.f.: Series reversion of x/P(x)^3 where P(x) is the g.f. for Partition numbers (A000041).

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A198954 Expansion of the rotational partition function for a heteronuclear diatomic molecule.

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Maple

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A319933 A(n, k) = [x^k] DedekindEta(x)^n, square array read by descending antidiagonals, A(n, k) for n >= 0 and k >= 0.

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Julia

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Sage

A202394 Expansion of f(-x)^3 + 9 * x * f(-x^9)^3 in powers of x where f() is a Ramanujan theta function.

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A215597 Expansion of psi(-x) * f(-x)^3 in powers of x where psi(), f() are Ramanujan theta functions.

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A227317 Expansion of psi(x)^6 * phi(-x)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

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