A045831 -id:A045831 - cp's OEIS frontend

A005875 Theta series of simple cubic lattice; also number of ways of writing a nonnegative integer n as a sum of 3 squares (zero being allowed).

1, 6, 12, 8, 6, 24, 24, 0, 12, 30, 24, 24, 8, 24, 48, 0, 6, 48, 36, 24, 24, 48, 24, 0, 24, 30, 72, 32, 0, 72, 48, 0, 12, 48, 48, 48, 30, 24, 72, 0, 24, 96, 48, 24, 24, 72, 48, 0, 8, 54, 84, 48, 24, 72, 96, 0, 48, 48, 24, 72, 0, 72, 96, 0, 6, 96, 96, 24, 48, 96, 48, 0, 36, 48, 120

Offset: 0

N. J. A. Sloane

Number of ordered triples (i, j, k) of integers such that n = i^2 + j^2 + k^2.
The Madelung Coulomb energy for alternating unit charges in the simple cubic lattice is Sum_{n>=1} (-1)^n*a(n)/sqrt(n) = -A085469. - R. J. Mathar, Apr 29 2006
a(A004215(k))=0 for k=1,2,3,... but no other elements of {a(n)} are zero. - Graeme McRae, Jan 15 2007

			Order and signs are taken into account: a(1) = 6 from 1 = (+-1)^2 + 0^2 + 0^2, a(2) = 12 from 2 = (+-1)^2 + (+-1)^2 + 0^2; a(3) = 8 from 3 = (+-1)^2 + (+-1)^2 + (+-1)^2, etc.
G.f. =  1 + 6*q + 12*q^2 + 8*q^3 + 6*q^4 + 24*q^5 + 24*q^6 + 12*q^8 + 30*q^9 + 24*q^10 + ...

H. Cohen, Number Theory, Vol. 1: Tools and Diophantine Equations, Springer-Verlag, 2007, p. 317.
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 107.
H. Davenport, The Higher Arithmetic. Cambridge Univ. Press, 7th ed., 1999, Chapter V.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 3, p. 109.
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 54.
L. Kronecker, Crelle, Vol. LVII (1860), p. 248; Werke, Vol. IV, p. 188.
C. J. Moreno and S. S. Wagstaff, Jr., Sums of Squares of Integers, Chapman and Hall, 2006, p. 43.
T. Nagell, Introduction to Number Theory, Wiley, 1951, p. 194.
W. Sierpiński, 1925. Teorja Liczb. pp. 1-410 (p.61).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
H. J. S. Smith, Report on the Theory of Numbers, reprinted in Vol. 1 of his Collected Math. Papers, Chelsea, NY, 1979, see p. 338, Eq. (B').

T. D. Noe, Table of n, a(n) for n = 0..10000
George E. Andrews, The Bhargava-Adiga Summation and Partitions, 2016. See Lemma 2.1.
P. T. Bateman, On the representations of a number as the sum of three squares, Trans. Amer. Math. Soc. 71 (1951), 70-101.
S. Bhargava and C. Adiga, A basic bilateral series summation formula and its applications, Integral Transforms and Special Functions, 2 (1994), 165-184.
J. M. Borwein, K-K S. Choi, On Dirichlet series for sums of squares, Raman. J. 7 (2003) 95-127
S. K. K. Choi, A. V. Kumchev, and R. Osburn, On sums of three squares, arXiv:math/0502007 [math.NT], 2005.
M. Doring, J. Haidenbauer, U.-G. Meissner, and A. Rusetsky, Dynamical coupled-channel approaches on a momentum lattice, arXiv:1108.0676 [hep-lat], 2011.
J. A. Ewell, Recursive determination of the enumerator for sums of three squares, Internat. J. Math. and Math. Sci, 24 (2000), 529-532.
O. Fraser and B. Gordon, On representing a square as the sum of three squares, Amer. Math. Monthly, 76 (1969), 922-923.
M. D. Hirschhorn and J. A. Sellers, On Representations of a Number as a Sum of Three Squares, Discrete Mathematics 199 (1999), 85-101.
M. D. Hirschhorn and J. A. Sellers, On Representations Of A Number As A Sum Of Three Squares
A. Martinez Torres, L. R. Dai, C. Koren, D. Jido, and E. Oset, The KD, eta D_s interaction in finite volume and the D_{s^*0}(2317) resonance, arXiv:1109.0396 [hep-lat], 2011.
R. J. Mathar, Hierarchical Subdivision of the Simple Cubic Lattice, arXiv:1309.3705 [math.MG], 2013.
S. C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., 6 (2002), 7-149.
J. L. Mordell, The Representation Of Integers By Three Positive Squares, Michigan Math. J. 7(3): 289-290 (1960).
Eric T. Mortenson, A Kronecker-type identity and the representations of a number as a sum of three squares, arXiv:1702.01627 [math.NT], 2017.
G. Nebe and N. J. A. Sloane, Home page for this lattice
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Theta Series
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for sequences related to sums of squares

Row d=3 of A122141 and of A319574, 3rd column of A286815.
Cf. A074590 (primitive solutions), A117609 (partial sums), A004215 (positions of zeros).
Analog for 4 squares: A000118.
x^2+y^2+k*z^2: A005875, A014455, A034933, A169783, A169784.
Cf. A000164, A004013, A004015, A008443, A045828, A045831, A045834, A213384.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

# JacobiTheta3 is defined in A000122.
A005875List(len) = JacobiTheta3(len, 3)
A005875List(75) |> println # Peter Luschny, Mar 12 2018

Basis( ModularForms( Gamma1(4), 3/2), 75) [1]; /* Michael Somos, Jun 25 2014 */

(sum(x^(m^2),m=-10..10))^3; seq(coeff(%,x,n), n=0..50);
Alternative:
A005875list := proc(len) series(JacobiTheta3(0, x)^3, x, len+1);
seq(coeff(%, x, j), j=0..len-1) end: A005875list(75); # Peter Luschny, Oct 02 2018

SquaresR[3,Range[0,80]] (* Harvey P. Dale, Jul 21 2011 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^3, {q, 0, n}]; (* Michael Somos, Jun 25 2014 *)
a[ n_] := Length @ FindInstance[ n == x^2 + y^2 + z^2, {x, y, z}, Integers, 10^9]; (* Michael Somos, May 21 2015 *)
QP = QPochhammer; CoefficientList[(QP[q^2]^5/(QP[q]*QP[q^4])^2)^3 + O[q]^80, q] (* Jean-François Alcover, Nov 24 2015 *)

{a(n) = if( n<0, 0, polcoeff( sum( k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n))^3, n))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 / (eta(x + A) * eta(x^4 + A))^2)^3, n))}; /* Michael Somos, Jun 03 2012 */

{a(n) = my(G); if( n<0, 0, G = [ 1, 0, 0; 0, 1, 0; 0, 0, 1]; polcoeff( 1 + 2 * x * Ser( qfrep( G, n)), n))}; /* Michael Somos, May 21 2015 */

# uses Python code for A004018
from math import isqrt
def A005875(n): return A004018(n)+(sum(A004018(n-k**2) for k in range(1,isqrt(n)+1))<<1) # Chai Wah Wu, Jun 21 2024

Q = DiagonalQuadraticForm(ZZ, [1]*3)
Q.representation_number_list(75) # Peter Luschny, Jun 20 2014

A number n is representable as the sum of 3 squares iff n is not of the form 4^a (8k+7) (cf. A000378).
There is a classical formula (essentially due to Gauss):
For sums of 3 squares r_3(n): write (uniquely) -n=D(2^vf)^2, with D<0 fundamental discriminant, f odd, v>=-1. Then r_3(n) = 12L((D/.),0)(1-(D/2)) Sum_{d | f} mu(d)(D/d)sigma(f/d).
Here mu is the Moebius function, (D/2) and (D/d) are Kronecker-Legendre symbols, sigma is the sum of divisors function, L((D/.),0)=h(D)/(w(D)/2) is the value at 0 of the L function of the quadratic character (D/.), equal to the class number h(D) divided by 2 or 3 in the special cases D=-4 and -3. - Henri Cohen (Henri.Cohen(AT)math.u-bordeaux1.fr), May 12 2010
a(n) = 3*T(n) if n == 1,2,5,6 mod 8, = 2*T(n) if n == 3 mod 8, = 0 if n == 7 mod 8 and = a(n/4) if n == 0 mod 4, where T(n) = A117726(n). [Moreno-Wagstaff].
"If 12E(n) is the number of representations of n as a sum of three squares, then E(n) = 2F(n) - G(n) where G(n) = number of classes of determinant -n, F(n) = number of uneven classes." - Dickson, quoting Kronecker. [Cf. A117726.]
a(n) = Sum_{d^2|n} b(n/d^2), where b() = A074590() gives the number of primitive solutions.
Expansion of phi(q)^3 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Oct 25 2006.
Euler transform of period 4 sequence [ 6, -9, 6, -3, ...]. - Michael Somos, Oct 25 2006
G.f.: (Sum_{k in Z} x^(k^2))^3.
a(8*n + 7) = 0. a(4*n) = a(n).
a(n) = A004015(2*n) = A014455(2*n) = A004013(4*n) = A169783(4*n). a(4*n + 1) = 6 * A045834(n). a(8*n + 3) = 8 * A008443(n). a(8*n + 5) = 24 * A045831(n). - Michael Somos, Jun 03 2012
a(4*n + 2) = 12 * A045828(n). - Michael Somos, Sep 03 2014
a(n) = (-1)^n * A213384(n). - Michael Somos, May 21 2015
a(n) = (6/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017
a(n) = A004018(n) + 2*Sum_{k=1..floor(sqrt(n))} A004018(n - k^2). - Daniel Suteu, Aug 27 2021
Convolution cube of A000122. Convolution of A004018 and A000122. - R. J. Mathar, Aug 03 2025

More terms from James Sellers, Aug 22 2000

A033687 Theta series of hexagonal lattice A_2 with respect to deep hole divided by 3.

Original entry on oeis.org

1, 1, 2, 0, 2, 1, 2, 0, 1, 2, 2, 0, 2, 0, 2, 0, 3, 2, 0, 0, 2, 1, 2, 0, 2, 2, 2, 0, 0, 0, 4, 0, 2, 1, 2, 0, 2, 2, 0, 0, 1, 2, 2, 0, 4, 0, 2, 0, 0, 2, 2, 0, 2, 0, 2, 0, 3, 2, 2, 0, 2, 0, 0, 0, 2, 3, 2, 0, 0, 2, 2, 0, 4, 0, 2, 0, 2, 0, 0, 0, 2, 2, 4, 0, 0, 1, 4, 0, 0, 2, 2, 0, 2, 0, 2, 0, 1, 2, 0, 0, 4, 2, 2, 0, 2

Offset: 0

N. J. A. Sloane

nonn

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
a(n)=0 if and only if A000731(n)=0 (see the Han-Ono paper). - Emeric Deutsch, May 16 2008
Number of 3-core partitions of n (denoted c_3(n) in Granville and Ono, p. 340). - Brian Hopkins, May 13 2008
Denoted by g_1(q) in Cynk and Hulek in Remark 3.4 on page 12 (but not explicitly listed).
This is a member of an infinite family of integer weight modular forms. g_1 = A033687, g_2 = A030206, g_3 = A130539, g_4 = A000731. - Michael Somos, Aug 24 2012
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + x + 2*x^2 + 2*x^4 + x^5 + 2*x^6 + x^8 + 2*x^9 + 2*x^10 + 2*x^12 + 2*x^14 + ...
G.f. = q + q^4 + 2*q^7 + 2*q^13 + q^16 + 2*q^19 + q^25 + 2*q^28 + 2*q^31 + 2*q^37 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 111.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 79, Eq. (32.35) and (32.351).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Johnathan M. Borwein and Peter B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701. MR1010408 (91e:33012) see page 697.
Slawomir Cynk and Klaus Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds, arXiv:math/0509424 [math.AG], 2005-2006.
Andrew Granville and Ken Ono, Defect Zero p-blocks for Finite Simple Groups, Transactions of the American Mathematical Society, Vol. 348 (1996), pp. 331-347.
Guo-Niu Han and Ken Ono, Hook Lengths and 3-Cores, Ann. Comb. (2011) Vol. 15, 305-312. See also arXiv:0805.2461, [math.NT], 2008.
Jeremy Lovejoy and Olivier Mallet, n-color overpartitions, twisted divisor functions, and Rogers-Ramanujan identities, South East Asian J. Math. Math. Sci., 6 (2008), 23-36. [From _Jeremy Lovejoy_, Jun 12 2009]
Gabriele Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000731, A002324, A005882, A006938, A033685.

Cf. A045831, A053723, A081622.

Basis( ModularForms( Gamma1(9), 1), 316) [2]; /* Michael Somos, May 06 2015 */

a[ n_] := If[ n < 0, 0, DivisorSum[ 3 n + 1, KroneckerSymbol[ -3, #] &]]; (* Michael Somos, Sep 23 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^3 / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Sep 01 2015 *)

{a(n) = if( n<0, 0, sumdiv( 3*n + 1, d, kronecker( -3, d)))}; /* Michael Somos, Nov 03 2005 */

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

{a(n) = my(A, p, e); if( n<0, 0, A = factor( 3*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 0, p%6==1, e+1, 1-e%2)))}; /* Michael Somos, May 06 2015 */

Euler transform of period 3 sequence [1, 1, -2, ...].
Expansion of q^(-1/3) * eta(q^3)^3 / eta(q) in powers of q.
a(4*n + 1) = a(n). - Michael Somos, Dec 06 2004
a(n) = b(3*n + 1) where b(n) is multiplicative and b(p^e) = 0^e if p = 3, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1+(-1)^e)/2 if p == 2, 5 (mod 6). - Michael Somos, May 20 2005
Given g.f. A(x), B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^2*w - 2*u*w^2 - v^3. - Michael Somos, Dec 06 2004
Given g.f. A(x), B(q)= q * A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1*u3^2 + u1*u6^2 - u1*u3*u6 - u2^2*u3. - Michael Somos, May 20 2005
Given g.f. A(x), B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u2*u3^2 + 2*u2*u3*u6 + 4*u2*u6^2 - u1^2*u6. - Michael Somos, May 20 2005
G.f.: Product_{k>0} (1 - x^(3*k))^3 / (1 - x^k).
G.f.: Sum_{k in Z} x^k / (1 - x^(3*k + 1)) = Sum_{k in Z} x^k / (1 - x^(6*k + 2)). - Michael Somos, Nov 03 2005
Expansion of q^(-1) * c(q^3) / 3 = q^(-1) * (a(q) - b(q)) / 9 in powers of q^3 where a(), b(), c() are cubic AGM theta functions. - Michael Somos, Dec 25 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (27 t)) = 3^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A005928.
a(n) = Sum_{d|3n+1} LegendreSymbol{d,3} - Brian Hopkins, May 13 2008
q-series for a(n): Sum_{n >= 0} q^(n^2+n)(1-q)(1-q^2)...(1-q^n)/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n+1))). [From Jeremy Lovejoy, Jun 12 2009]
a(n) = A002324(3*n + 1). 3*a(n) = A005882(n) = A033685(3*n + 1). - Michael Somos, Apr 04 2003
G.f.: (2 * psi(x^2) * f(x^2, x^4) + phi(x) * f(x^1, x^5)) / 3 where phi(), psi() are Ramanujan theta functions and f(, ) is Ramanujan's general theta function. - Michael Somos, Sep 07 2018
Sum_{k=1..n} a(k) ~ 2*Pi*n/3^(3/2). - Vaclav Kotesovec, Dec 17 2022

A259825 a(n) = 12*H(n) where H() is the Hurwitz class number.

Original entry on oeis.org

-1, 0, 0, 4, 6, 0, 0, 12, 12, 0, 0, 12, 16, 0, 0, 24, 18, 0, 0, 12, 24, 0, 0, 36, 24, 0, 0, 16, 24, 0, 0, 36, 36, 0, 0, 24, 30, 0, 0, 48, 24, 0, 0, 12, 48, 0, 0, 60, 40, 0, 0, 24, 24, 0, 0, 48, 48, 0, 0, 36, 48, 0, 0, 60, 42, 0, 0, 12, 48, 0, 0, 84, 36, 0, 0

Offset: 0

Michael Somos, Jul 05 2015

sign

Coefficients of q-expansion of Eisenstein series G_{3/2}(tau) multiplied by 12. - N. J. A. Sloane, Mar 16 2019

			G.f. = -1 + 4*x^3 + 6*x^4 + 12*x^7 + 12*x^8 + 12*x^11 + 16*x^12 + 24*x^15 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Kathrin Bringmann and Jeremy Lovejoy, Overpartitions and class numbers of binary quadratic forms, arXiv:0712.0631 [math.NT], 2007. See page 5, equation (1.12).
Archer Clayton, Helen Dai, Tianyu Ni, Erick Ross, Hui Xue, and Jake Zummo, Non-repetition of second coefficients of Hecke polynomials, arXiv:2411.18419 [math.NT], 2024. See p. 21.
Wikipedia, Hurwitz class number.
Don Zagier, Modular Forms of One Variable, Notes based on a course given in Utrecht, 1991. See page 50.

Cf. A004024, A005869, A005876, A005877, A005878, A005884, A005886, A008443.
Cf. A045828, A045831, A045834, A058305, A058306, A130695, A181648, A185220.
Cf. A188569, A213022, A213617, A213624, A213625, A213627, A259827.
See also A306934-A306937.

terms = 100; gf[m_] := With[{r = Range[-m, m]}, -2 Sum[(-1)^k*x^(k^2 + k)/(1 + (-x)^k)^2, {k, r}]/EllipticTheta[3, 0, x] - 2 Sum[(-1)^k*x^(k^2 + 2 k)/(1 + x^(2 k))^2, {k, r}]/EllipticTheta[3, 0, -x]]; gf[terms // Sqrt // Ceiling] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Apr 02 2017 *)
a[ n_] := If[ n<1, -Boole[n==0], With[{m = Floor[(-1 + Sqrt[1 + 4*n])/2]}, -2*SeriesCoefficient[ Sum[(-1)^k*x^(k^2 + k)/(1 + (-x)^k)^2, {k, -m-1,m}] / EllipticTheta[3, 0, x] + Sum[(-1)^k*x^(k^2 + 2*k)/(1 + x^(2*k))^2, {k, -m-2,m}]/ EllipticTheta[3, 0, -x], {x, 0, n}]]]; (* Michael Somos, Feb 04 2022 *)

PARI
```
{a(n) = 12 * qfbhclassno(n)};
```

{a(n) = my(D, f); 12 * if( n<1, (n==0)/-12, [D, f] = core(-n, 1); if( D%4>1 && !(f%2), D*=4; f/=2); if( D%4<2, qfbclassno(D) / max(1, D+6), 0) * sumdiv(f, d, moebius(d) * kronecker(D, d) * sigma(f/d)))};

a(n) = 12 * A058305(n) / A058306(n). a(4*n + 1) = a(4*n + 2) = 0. a(3*n + 4) = 6 * A259827(n).
a(4*n + 3) = 4 * A130695(n). a(8*n + 3) = A005886(n) = 2 * A005869(n) = 4 * A008443(n). a(12*n + 7) = 12 * A259655(n).
a(16*n + 4) = 6 * A045834(n) = 3 * A005876(n). a(16*n + 8) = 12 * A045828(n) = 6 * A005884(n) = 3 * A005877(n).
a(24*n + 3) = 4 * A213627(n). a(24*n + 7) = 12 * A185220(n). a(24*n + 11) = 12 * A213617(n). a(24*n + 19) = 12 * A181648(n). a(24*n + 23) = 12 * A188569(n+1).
a(32*n + 4) = 6 * A213022(n). a(32*n + 8) = 12 * A213625(n). a(32*n + 12) = 16 * A008443(n) = 8 * A005869(n) = 4 * A005886(n) = 2 * A005878(n). a(32*n + 20) = 24 * A045831(n) = 6 * A004024(n). a(32*n + 24) = 24 * A213624(n).
G.f.: -2 * (Sum_{k in Z} (-1)^k * x^(k*k + k) / (1 + (-x)^k)^2) / (Sum_{k in Z} x^k^2) - 2 * (Sum_{k in Z} (-1)^k * x^(k^2 + 2*k) / (1 + x^(2*k))^2) / (Sum_{k in Z} (-x)^k^2).
a(n) >= 0 if n > 0. - Michael Somos, Feb 04 2022

A175595 Square array A(n,t), n>=0, t>=0, read by antidiagonals: A(n,t) is the number of t-core partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 1, 0, 3, 1, 1, 0, 0, 5, 1, 1, 2, 1, 0, 7, 1, 1, 2, 0, 0, 0, 11, 1, 1, 2, 3, 2, 0, 0, 15, 1, 1, 2, 3, 1, 1, 1, 0, 22, 1, 1, 2, 3, 5, 3, 2, 0, 0, 30, 1, 1, 2, 3, 5, 2, 3, 0, 0, 0, 42, 1, 1, 2, 3, 5, 7, 6, 3, 1, 0, 0, 56, 1, 1, 2, 3, 5, 7, 5, 5, 4, 2, 1, 0, 77, 1, 1, 2, 3, 5, 7, 11, 9, 7, 4, 2, 0, 0, 101

Offset: 0

Alois P. Heinz, Dec 03 2010

A partition of n is a t-core partition if none of the hook numbers associated to the Ferrers-Young diagram is a multiple of t. See Chen reference for definitions.

			A(4,3) = 2, because there are 2 partitions of 4 such that no hook number is a multiple of 3:
   (1)  2 | 4 1
       +1 | 2
       +1 | 1
   -------+-----
   (2)  3 | 4 2 1
       +1 | 1
Square array A(n,t) begins:
   1,  1,  1,  1,  1,  1,  1,  1,  ...
   1,  0,  1,  1,  1,  1,  1,  1,  ...
   2,  0,  0,  2,  2,  2,  2,  2,  ...
   3,  0,  1,  0,  3,  3,  3,  3,  ...
   5,  0,  0,  2,  1,  5,  5,  5,  ...
   7,  0,  0,  1,  3,  2,  7,  7,  ...
  11,  0,  1,  2,  3,  6,  5, 11,  ...
  15,  0,  0,  0,  3,  5,  9,  8,  ...

Garvan, F. G., A number-theoretic crank associated with open bosonic strings. In Number Theory and Cryptography (Sydney, 1989), 221-226, London Math. Soc. Lecture Note Ser., 154, Cambridge Univ. Press, Cambridge, 1990.
James, Gordon; and Kerber, Adalbert, The Representation Theory of the Symmetric Group. Addison-Wesley Publishing Co., Reading, Mass., 1981.

Alois P. Heinz, Antidiagonals n = 0..140, flattened
G. E. Andrews and F. Garvan, Dyson's crank of a partition, Bull. Amer. Math. Soc., 18 (1988), 167-171.
A. O. L. Atkins and F. G. Garvan, Relations between the ranks and cranks of partitions, arXiv:math/0208050 [math.NT], 2002.
A. O. L. Atkins and F. G. Garvan, Relations between the ranks and cranks of partitions, Rankin memorial issues. Ramanujan J. 7 (2003), 343-366.
Shichao Chen, Arithmetical properties of the number of t-core partitions, The Ramanujan Journal, 18 (2007), no. 1, 103-112, DOI: 10.1007/s11139-007-9045-5.
F. G. Garvan, The crank of partitions mod 8, 9 and 10, Trans. Amer. Math. Soc. 322 (1990), 79-94.
F. G. Garvan, Some congruences for partitions that are p-cores, Proc. London Math. Soc. 66 (1993), 449-478.
F. G. Garvan, More cranks and t-cores, Bull. Austral. Math. Soc. 63 (2001), 379-391.
F. G. Garvan, D. Kim and D. Stanton, Cranks and t-cores, Inventiones Math. 101 (1990) 1-17.
Andrew Granville and Ken Ono, Defect Zero p-blocks for Finite Simple Groups, Transactions of the American Mathematical Society, Vol. 348 (1996), pp. 331-347.
Ben Kane, Sums of Triangular Numbers and t-Core Partitions, Journal of Combinatorics and Number Theory, 1 (2009), no.1, 59-64.
B. Kim, On inequalities and linear relations for 7-core partitions, Discrete Math., 310 (2010), 861-868.
N. J. A. Sloane, Transforms.

Columns t=0-12 give A000041, A000007, A010054, A033687, A045831, A053723, A081622, A053724, A182803, A182804, A182805, A053691, A192061.
Rows n=0-1 give A000012, A060576.
Diagonal gives A000094(n+1) for n>0.
Upper diagonal gives A000041.
Lower diagonal (conjectured) gives A086642 for n>0.

with(numtheory):
A:= proc(n, t) option remember; `if`(n=0, 1,
      add(add(`if`(t=0 or irem(d, t)=0, d-d*t, d),
              d=divisors(j))*A(n-j, t), j=1..n)/n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);
(From N. J. A. Sloane, Jun 21 2011: to get M terms of the series for t-core partitions:)
M:=60;
f:=proc(t) global M; local q,i,t1;
t1:=1;
for i from 1 to M+1 do
t1:=series(t1*(1-q^(i*t))^t,q,M);
t1:=series(t1/(1-q^i),q,M);
od;
t1;
end;
# then for example seriestolist(f(5));

n = 13; f[t_] = (1-x^(t*k))^t/(1-x^k); f[0] = 1/(1-x^k);
s[t_] := CoefficientList[ Series[ Product[ f[t], {k, 1, n}], {x, 0, n}], x]; m = Table[ PadRight[ s[t], n+1], {t, 0, n}]; Flatten[ Table[ m[[j+1-k, k]], {j, n+1}, {k, j}]] (* Jean-François Alcover, Jul 25 2011, after g.f. *)

G.f. of column t: Product_{i>=1} (1-x^(t*i))^t/(1-x^i).

Column t is the Euler transform of period t sequence [1, .., 1, 1-t, ..].

Additional references from N. J. A. Sloane, Jun 21 2011

A004024 Theta series of b.c.c. lattice with respect to deep hole.

Original entry on oeis.org

4, 4, 8, 12, 4, 12, 12, 12, 16, 16, 8, 8, 28, 12, 20, 24, 8, 16, 28, 12, 16, 28, 20, 32, 20, 16, 16, 32, 20, 24, 28, 8, 36, 44, 12, 32, 36, 16, 24, 20, 28, 20, 56, 28, 16, 40, 20, 40, 44, 12, 36, 40, 20, 32, 40, 16, 24, 60, 32, 36, 40, 24, 32, 60, 24, 40, 24, 20, 60, 36, 24, 32, 56, 32

Offset: 0

N. J. A. Sloane

Ono and Skinner, Ann. Math., 147 (1998), 453-470.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane and B. K. Teo, Theta series and magic numbers for close-packed spherical clusters, J. Chem. Phys. 83 (1985) 6520-6534.

T. D. Noe, Table of n, a(n) for n = 0..1000
G. Nebe and N. J. A. Sloane, Home page for this lattice
Index entries for sequences related to b.c.c. lattice

Equals 4*A045831.

max = 73; 4*CoefficientList[ Series[ Product[ (1-q^(4k))^4 / (1-q^k), {k, 1, max}], {q, 0, max}], q] (* Jean-François Alcover, Feb 10 2012, after A045831 *)
terms = 74; QP = QPochhammer; s = 4 QP[z^4]^4/QP[z] + O[z]^terms; CoefficientList[s, z] (* Jean-François Alcover, Jul 07 2017 *)

4*eta(32z)^4/eta(8z) = 4*Sum q^(x^2+2y^2+2z^2), x, y, z >= 1 and odd.

A033717 Number of integer solutions to the equation x^2 + 2y^2 + 4z^2 = n.

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 8, 8, 6, 6, 8, 4, 8, 12, 0, 8, 12, 8, 10, 12, 8, 8, 24, 8, 8, 14, 8, 16, 16, 4, 0, 16, 6, 16, 16, 8, 12, 20, 24, 8, 24, 8, 16, 20, 8, 20, 0, 16, 24, 18, 10, 8, 24, 12, 32, 24, 0, 16, 24, 12, 16, 20, 0, 24, 12, 8, 16, 28, 16, 16, 48, 8, 30, 32, 8, 20, 24, 16, 0, 16, 24, 18

Offset: 0

N. J. A. Sloane

nonn

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

			G.f. = 1 + 2*q + 2*q^2 + 4*q^3 + 4*q^4 + 4*q^5 + 8*q^6 + 8*q^7 + 6*q^8 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p 102 eq 9.

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. A004015, A008443, A014455, A033763, A045826, A045828, A045831, A045834, A213022, A213622, A213624, A213625, A246811.

A := Basis( ModularForms( Gamma1(16), 3/2), 82); A[1] + 2*A[2] + 2*A[3] + 4*A[4] + 4*A[5] + 4*A[6] + 8*A[7] + 8*A[8] + 6*A[9] + 8*A[10] + 4*A[11]; /* Michael Somos, Sep 03 2014 */

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] EllipticTheta[ 3, 0, q^4], {q, 0, n}]; (* Michael Somos, Sep 03 2014 *)

{a(n) = my(G); if( n<0, 0, G = [1, 0, 0; 0, 2, 0; 0, 0, 4]; polcoeff( 1 + 2 * x * Ser(qfrep( G, n)), n))}; /* Michael Somos, Sep 03 2014 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^4 + A) * eta(x^8 + A)^3 / (eta(x + A)^2 * eta(x^16 + A)^2), n))}; /* Michael Somos, Sep 03 2014 */

Expansion of phi(q) * phi(q^2) * phi(q^4) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Sep 03 2014
Euler transform of period 16 sequence [2, -1, 2, -2, 2, -1, 2, -5, 2, -1, 2, -2, 2, -1, 2, -3, ...]. - Michael Somos, Sep 03 2014
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 8 (t/i)^(3/2) f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 03 2014
a(2*n + 1) = 2 * A045828(n). a(4*n) = A014455(n). a(4*n + 1) = 2 * A213625(n). a(4*n + 2) = 2 * A246811(n). a(4*n + 3) = 4 * A213624(n). - Michael Somos, Sep 03 2014
a(8*n) = A005875(n). a(8*n + 1) = 2 * A213622(n). a(8*n + 2) = 2 * A045834(n). a(8*n + 7) = 8 * A033763(n). - Michael Somos, Sep 03 2014
a(16*n) = A004015(n). a(16*n + 2) = 2 * A213022(n). a(16*n + 6) = 8 *
A008443(n). a(16*n + 8) = 2 * A045826(n). a(16*n + 10) = 8 * A045831(n). a(16*n + 14) = 0. - Michael Somos, Sep 03 2014
G.f.: theta_3(q) * theta_3(q^2) * theta_3(q^4).

A081622 Number of 6-core partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 5, 9, 10, 12, 12, 14, 20, 20, 21, 23, 24, 24, 32, 29, 35, 36, 44, 47, 38, 47, 49, 52, 55, 58, 59, 64, 66, 71, 70, 78, 79, 88, 87, 90, 85, 87, 111, 104, 102, 107, 112, 113, 121, 113, 130, 130, 148, 153, 132, 147, 149, 156, 162, 149, 167, 160, 178, 180

Offset: 0

Michael Somos, Mar 24 2003

Euler transform of period 6 sequence [ 1, 1, 1, 1, 1, -5, ...].

Expansion of q^(-35/24) * eta(q^6)^6 / eta(q) in powers of q.

			1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 5*x^6 + 9*x^7 + 10*x^8 + 12*x^9 + ...
q^35 + q^59 + 2*q^83 + 3*q^107 + 5*q^131 + 7*q^155 + 5*q^179 + 9*q^203 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
F. Garvan, D. Kim and D. Stanton, Cranks and t-cores, Inventiones Math. 101 (1990) 1-17.

Cf. A010054, A033687, A045831, A053723, A053724.

{a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^(6*k) + x * O(x^n))^6 / (1 - x^k)), n))}

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^6 / eta(x + A), n))}

G.f.: Product_{k>0} (1 - x^(6*k))^6 / (1 - x^k).

A182805 Number of 10-core partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 32, 46, 57, 71, 85, 106, 121, 147, 165, 190, 242, 267, 302, 350, 400, 443, 511, 565, 638, 715, 774, 852, 964, 1038, 1135, 1253, 1372, 1482, 1650, 1785, 1878, 2098, 2234, 2411, 2625, 2819, 2963, 3249, 3393, 3600, 4004, 4181

Offset: 0

Alois P. Heinz, Dec 03 2010

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

10th column of A175595.

Cf. A033687, A045831, A053723, A081622, A053724, A182803, A182804.

with(numtheory): A:= proc(n, t) option remember; local d, j; `if`(n=0, 1, add(add(`if`(t=0 or irem(d, t)=0, d-d*t, d), d=divisors(j)) *A(n-j, t), j=1..n)/n) end: seq(A(n,10), n=0..50);

A[n_, t_] := A[n, t] = Module[{d, j}, If[n == 0, 1, Sum[Sum[If[t == 0 || Mod[d, t] == 0, d - d t, d], {d, Divisors[j]}] A[n - j, t], {j, 1, n}]/n]];
Table[A[n, 10], {n, 0, 50}] (* Jean-François Alcover, Dec 06 2020, after Alois P. Heinz *)

G.f.: Product_{i>=1} (1-x^(10*i))^10/(1-x^i).

Euler transform of period 10 sequence [1,1,1,1,1,1,1,1,1,-9, .. ].

A246811 Expansion of phi(x)^2 * psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 4, 4, 0, 5, 12, 4, 0, 8, 12, 8, 0, 5, 16, 12, 0, 8, 24, 4, 0, 16, 12, 12, 0, 9, 24, 12, 0, 8, 36, 12, 0, 16, 12, 16, 0, 8, 28, 16, 0, 17, 36, 8, 0, 24, 24, 8, 0, 8, 36, 28, 0, 16, 36, 12, 0, 16, 24, 20, 0, 13, 24, 24, 0, 24, 60, 8, 0, 16, 36, 16, 0, 16, 28

Offset: 0

Michael Somos, Sep 03 2014

nonn

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

			G.f. = 1 + 4*x + 4*x^2 + 5*x^4 + 12*x^5 + 4*x^6 + 8*x^8 + 12*x^9 + ...
G.f. = q + 4*q^3 + 4*q^5 + 5*q^9 + 12*q^11 + 4*q^13 + 8*q^17 + 12*q^19 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A008443, A033717, A045831, A045834, A213022.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x]^2 EllipticTheta[ 2, 0, x^2] / (2 x^(1/2)), {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(1/2)]^4 / (16 x^(1/2) EllipticTheta[ 3, 0, x^2]), {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^10 * eta(x^8 + A)^2 / (eta(x + A)^4 * eta(x^4 + A)^5), n))};

Expansion of psi(x)^4 / phi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of q^(-1/2) * eta(q^2)^10 * eta(q^8)^2 / (eta(q)^4 * eta(q^4)^5) in powers of q.
Euler transform of period 8 sequence [4, -6, 4, -1, 4, -6, 4, -3, ...].
2 * a(n) = A033717(4*n + 2). a(2*n) = A045834(n). a(4*n) = A213022(n). a(4*n + 1) = 4 * A008443(n). a(4*n + 2) = 4 * A045831(n). a(4*n + 3) = 0.

A246954 Expansion of phi(-x) * psi(-x)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -4, 5, -4, 8, -8, 5, -12, 8, -4, 16, -12, 9, -12, 8, -12, 16, -16, 8, -16, 17, -8, 24, -8, 8, -28, 16, -12, 16, -20, 13, -24, 24, -8, 16, -16, 16, -28, 24, -12, 32, -16, 13, -28, 8, -20, 32, -32, 8, -20, 24, -16, 40, -16, 16, -32, 25, -20, 24, -24, 24, -28

Offset: 0

Michael Somos, Sep 08 2014

sign

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

			G.f. = 1 - 4*x + 5*x^2 - 4*x^3 + 8*x^4 - 8*x^5 + 5*x^6 - 12*x^7 + 8*x^8 + ...
G.f. = q - 4*q^5 + 5*q^9 - 4*q^13 + 8*q^17 - 8*q^21 + 5*q^25 - 12*q^29 + ...

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. A045831, A045834, A246953.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x]^2 EllipticTheta[ 2, 0, x]/(2 x^(1/4)), {x, 0, n}]; Table[a[n], {n, 0, 60}]

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^4 + A)^2 / eta(x^2 + A)^3, n))};

Expansion of phi(-x)^2 * psi(x^2) = psi(-x)^4 / psi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of q^(-1/4) * eta(q)^4 * eta(q^4)^2 / eta(q^2)^3 in powers of q.
Euler transform of period 4 sequence [ -4, -1, -4, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 16 * (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A246953.
G.f.: Product_{k>0} (1 - x^k)^3 * (1 + x^(2*k))^2 / (1 + x^k).
a(n) = (-1)^n * A045834(n). a(2*n) = A213022(n). a(2*n + 1) = - 4 * A045831(n).

cp's OEIS Frontend

A005875 Theta series of simple cubic lattice; also number of ways of writing a nonnegative integer n as a sum of 3 squares (zero being allowed).

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A033687 Theta series of hexagonal lattice A_2 with respect to deep hole divided by 3.

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A259825 a(n) = 12*H(n) where H() is the Hurwitz class number.

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A175595 Square array A(n,t), n>=0, t>=0, read by antidiagonals: A(n,t) is the number of t-core partitions of n.

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A004024 Theta series of b.c.c. lattice with respect to deep hole.

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A033717 Number of integer solutions to the equation x^2 + 2*y^2 + 4*z^2 = n.

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A033717 Number of integer solutions to the equation x^2 + 2y^2 + 4z^2 = n.