A007331 -id:A007331 - cp's OEIS frontend

A206622 G.f.: Product_{n>0} ( (1+x^n)/(1-x^n) )^(n^2).

1, 2, 10, 36, 118, 376, 1148, 3376, 9654, 26894, 73192, 195188, 510948, 1315048, 3332720, 8326448, 20529526, 49998884, 120379574, 286726340, 676057144, 1578880480, 3654180236, 8385122192, 19085029540, 43103203626, 96630606968, 215105226728, 475608824400

Offset: 0

Paul D. Hanna, Feb 10 2012

nonn

Compare g.f. to: Product_{n>0} (1+x^n)/(1-x^n) = exp( Sum_{n>=1} (sigma(2*n) - sigma(n))*x^n/n ) which equals 1/theta_4(x) = 1/(1 + 2*Sum_{n>=1} (-x)^(n^2)).
Convolution of A023871 and A027998. - Vaclav Kotesovec, Aug 19 2015
In general, if g.f. = Product_{k>=1} ((1 + x^k)/(1 - x^k))^(c2*k^2 + c1*k + c0) and c2>0, then a(n) ~ exp(Pi * 2^(5/4) * c2^(1/4) * n^(3/4) / 3 + 7*c1 * Zeta(3) * sqrt(n) / (Pi^2 * sqrt(2*c2)) + (c0*Pi / (2^(5/4) * c2^(1/4)) - 49*c1^2 * Zeta(3)^2 / (2^(5/4) * c2^(5/4) * Pi^5)) * n^(1/4) + 22411 * c1^3 * Zeta(3)^3 / (196 * c2^2 * Pi^8) - 7*c0*c1 * Zeta(3) / (4*c2 * Pi^2) - c2 * Zeta(3) / (4*Pi^2) + c1/12) * Pi^(c1/12) * c2^(1/8 + c0/8 + c1/48) / (A^c1 * 2^(15/8 + 11*c0/8 + 7*c1/48) * n^(5/8 + c0/8 + c1/48)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 08 2017
Let A(x) denote the g.f. and let m be an integer. Define a sequence by u(n) = [x^n] A(x)^(m*n). We conjecture that the supercongruence u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) holds for all positive integers n and r and all primes p >= 5. Cf. A380582. - Peter Bala, Jan 21 2025

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 36*x^3 + 118*x^4 + 376*x^5 + 1148*x^6 +...
where A(x) = (1+x)/(1-x) * (1+x^2)^4/(1-x^2)^4 * (1+x^3)^9/(1-x^3)^9 *...
Also, A(x) = Euler transform of [2,7,18,28,50,63,98,112,162,175,...]:
A(x) = 1/((1-x)^2*(1-x^2)^7*(1-x^3)^18*(1-x^4)^28*(1-x^5)^50*(1-x^6)^63*...).

Seiichi Manyama, Table of n, a(n) for n = 0..9042 (terms 0..1000 from Vaclav Kotesovec)
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. 23.

Cf. A156616, A206623, A206624, A001158 (sigma_3), A380582.

nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^(k^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 19 2015 *)

{a(n)=polcoeff(prod(m=1,n+1,((1+x^m)/(1-x^m+x*O(x^n)))^(m^2)),n)}

{a(n)=polcoeff(exp(sum(m=1, n, (sigma(2*m, 3)-sigma(m, 3))/4*x^m/m)+x*O(x^n)), n)}

{a(n)=local(InvEulerGF=x*(2+7*x+12*x^2+7*x^3+2*x^4)/(1-x^2+x*O(x^n))^3);polcoeff(1/prod(k=1,n,(1-x^k+x*O(x^n))^polcoeff(InvEulerGF,k)),n)}
for(n=0,35,print1(a(n),", "))

G.f.: exp( Sum_{n>=1} (sigma_3(2*n) - sigma_3(n))/4 * x^n/n ), where sigma_3(n) is the sum of cubes of divisors of n (A001158).
The inverse Euler transform has g.f.: x*(2 + 7*x + 12*x^2 + 7*x^3 + 2*x^4)/(1-x^2)^3.
a(n) ~ exp(2^(5/4)*Pi*n^(3/4)/3 - Zeta(3)/(4*Pi^2)) / (2^(15/8) * n^(5/8)), where Zeta(3) = A002117. - Vaclav Kotesovec, Aug 19 2015
a(0) = 1, a(n) = (2/n)*Sum_{k=1..n} A007331(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 30 2017

A076577 Sum of squares of divisors d of n such that n/d is odd.

Original entry on oeis.org

1, 4, 10, 16, 26, 40, 50, 64, 91, 104, 122, 160, 170, 200, 260, 256, 290, 364, 362, 416, 500, 488, 530, 640, 651, 680, 820, 800, 842, 1040, 962, 1024, 1220, 1160, 1300, 1456, 1370, 1448, 1700, 1664, 1682, 2000, 1850, 1952, 2366, 2120, 2210, 2560, 2451, 2604

Offset: 1

Vladeta Jovovic, Oct 19 2002

			G.f. = x + 4*x^2 + 10*x^3 + 16*x^4 + 26*x^5 + 40*x^6 + 50*x^7 + 64*x^8 + ...

Robert Israel, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher

Cf. A001227, A002131, A001157, A050999.

Glaisher's Delta'_i (i=0..12): A001227, A002131, A076577, A007331, A285989, A096960, A321817, A096961, A321818, A096962, A321819, A096963, A321820

a:= n -> mul(`if`(t[1]=2, 2^(2*t[2]),
     (t[1]^(2*(1+t[2]))-1)/(t[1]^2-1)),t=ifactors(n)[2]):
map(a, [$1..100]); # Robert Israel, Jul 05 2016

a[ n_] := If[ n < 1, 0, Sum[ d^2 Mod[ n/d, 2], {d, Divisors @ n}]]; (* Michael Somos, Jun 09 2014 *)
Table[CoefficientList[Series[-Log[Product[(x^k - 1)^k/(x^k + 1)^k, {k, 1, 80}]]/2, {x, 0, 80}], x][[n + 1]] n, {n, 1, 80}] (* Benedict W. J. Irwin, Jul 05 2016 *)
f[2, e_] := 4^e; f[p_, e_] := (p^(2*e + 2) - 1)/(p^2 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2020 *)

a(n) = sumdiv(n, d, d^2*((n/d) % 2)); \\ Michel Marcus, Jun 09 2014

G.f.: Sum_{m>0} m^2*x^m/(1-x^(2*m)). More generally, if b(n, k) is sum of k-th powers of divisors d of n such that n/d is odd then b(2n, k) = sigma_k(2n)-sigma_k(n), b(2n+1, k) = sigma_k(2n+1), where sigma_k(n) is sum of k-th powers of divisors of n. G.f. for b(n, k): Sum_{m>0} m^k*x^m/(1-x^(2*m)).
b(n, k) is multiplicative: b(2^e, k) = 2^(k*e), b(p^e, k) = (p^(ke+k)-1)/(p^k-1) for an odd prime p.
a(2*n) = sigma_2(2*n)-sigma_2(n), a(2*n+1) = sigma_2(2*n+1), where sigma_2(n) is sum of squares of divisors of n (cf. A001157).
b(n, k) = (sigma_k(2n)-sigma_k(n))/2^k. - Vladeta Jovovic, Oct 06 2003
Dirichlet g.f.: zeta(s)*(1-1/2^s)*zeta(s-2). - Geoffrey Critzer, Mar 28 2015
L.g.f.: -log(Product_{ k>0 } (x^k-1)^k/(x^k+1)^k)/2 = Sum_{ n>0 } (a(n)/n)*x^n. - Benedict W. J. Irwin, Jul 05 2016
Sum_{k=1..n} a(k) ~ 7*Zeta(3)*n^3 / 24. - Vaclav Kotesovec, Feb 08 2019

A226255 Number of ways of writing n as the sum of 11 triangular numbers.

Original entry on oeis.org

1, 11, 55, 176, 440, 957, 1848, 3245, 5412, 8580, 12892, 18888, 26895, 36916, 50160, 66935, 86658, 111870, 142582, 177320, 221100, 272690, 329065, 399102, 480040, 566808, 672969, 793760, 920326, 1074040, 1248412, 1425974, 1640595, 1882145, 2123385, 2418339, 2743928, 3062895, 3453978, 3880855

Offset: 0

N. J. A. Sloane, Jun 01 2013

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000
K. Ono, S. Robins and P. T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94.

Number of ways of writing n as a sum of k triangular numbers, for k=1,...: A010054, A008441, A008443, A008438, A008439, A008440, A226252, A007331, A226253, A226254, A226255, A014787, A014809.

G.f. is 11th power of g.f. for A010054.
a(0) = 1, a(n) = (11/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, May 06 2017
G.f.: exp(Sum_{k>=1} 11*(x^k/k)/(1 + x^k)). - Ilya Gutkovskiy, Jul 31 2017

A035016 Fourier coefficients of E_{0,4}.

Original entry on oeis.org

1, -16, 112, -448, 1136, -2016, 3136, -5504, 9328, -12112, 14112, -21312, 31808, -35168, 38528, -56448, 74864, -78624, 84784, -109760, 143136, -154112, 149184, -194688, 261184, -252016, 246176, -327040, 390784, -390240, 395136, -476672, 599152, -596736

Offset: 0

Barry Brent (barryb(AT)primenet.com)

E_{0,4} is unique normalized entire modular form of weight 4 for \Gamma_0(2) with a zero at zero. Also |a(n)| matches expansion of theta_3(z)^8 (A000143).

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

			G.f. = 1 - 16*q + 112*q^2 - 448*q^3 + 1136*q^4 - 2016*q^5 + 3136*q^6 - 5504*q^7 + ...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 77, Eq. (31.61).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Richard E. Borcherds, Automorphic forms with singularities on Grassmannians, arXiv:alg-geom/9609022, 1996-1997; Invent. Math. 132 (1998), 491-562.
B. Brent, Quadratic Minima and Modular Forms, Experimental Mathematics, v.7 no.3, 257-274.

Cf. A000143, A096727.

# JacobiTheta4 is defined in A002448.
A035016List(len) = JacobiTheta4(len, 8)
A035016List(34) |> println # Peter Luschny, Mar 12 2018

a_list := proc(len) series(JacobiTheta4(0,x)^8,x,len+1); seq(coeff(%,x,j),j=0..len) end: a_list(33); # Peter Luschny, Mar 14 2017

a[0] = 1; a[n_] := 16*Sum[(-1)^d*d^3, {d, Divisors[n]}]; Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Feb 06 2012, after Pari *)
QP = QPochhammer; s = QP[q]^16/QP[q^2]^8 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)

{a(n) = if( n<1, n==0, 16 * sumdiv( n, d, (-1)^d * d^3))};

{a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k) / (1 + x^k), 1 + x * O(x^n))^8, n))};

{a(n) = local(A); if( n<0, 0, A = x^n * O(x); polcoeff( (eta(x + A)^2 / eta(x^2 + A))^8, n))}; /* Michael Somos, Jan 11 2009 */

from sympy import divisors
def a(n): return 1 if n==0 else 16*sum((-1)**d*d**3 for d in divisors(n))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 24 2017

A = ModularForms( Gamma0(4), 4, prec=34) . basis(); A[0] - 16*A[1] + 112*A[2]; # Michael Somos, Jun 15 2014

a(0)=1; for n>0, a(n) = 16*sum_{0

G.f.: Product_{n>=1} ((1-q^n)/(1+q^n))^8 [Fine]

Expansion of phi(-q)^8 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Jun 15 2014

Expansion of eta(q)^16 / eta(q^2)^8 in powers of q.

Euler transform of period 2 sequence [ -16, -8, ...]. - Michael Somos, Apr 10 2005

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^3 + u*v * (u - 2*v + 16*w) - 16 * u*w^2. - Michael Somos, Apr 10 2005

G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = 256 (t / i)^4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A007331. - Michael Somos, Jan 11 2009

a(n) = (-1)^n * A000143(n).

Convolution square of A096727. - Michael Somos, Jun 15 2014

A096960 a(n) = Sum_{0

Original entry on oeis.org

1, 32, 244, 1024, 3126, 7808, 16808, 32768, 59293, 100032, 161052, 249856, 371294, 537856, 762744, 1048576, 1419858, 1897376, 2476100, 3201024, 4101152, 5153664, 6436344, 7995392, 9768751, 11881408, 14408200, 17211392, 20511150

Offset: 1

Ralf Stephan, Jul 18 2004

			G.f. = q + 32*q^2 + 244*q^3 + 1024*q^4 + 3126*q^5 + 7808*q^6 + 16808*q^7 + 32768*q^8 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..10000
H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Cf. A007331, A013664, A096961, A096962, A096963.

Basis( ModularForms( Gamma0(2), 6), 30) [2]; /* Michael Somos, Nov 30 2014 */

a[ n_] := If[ n < 1, 0, Sum[ d^5 Boole[ OddQ[ n/d]], {d, Divisors[ n]}]]; (* Michael Somos, Jun 04 2013 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^4 + EllipticTheta[ 2, 0, q]^4) EllipticTheta[ 2, 0, q^(1/2)]^8 / 256, {q, 0, n}]; (* Michael Somos, Jun 04 2013 *)

{a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d^5))};

ModularForms( Gamma0(2), 6, prec=33).gen(1).coefficients(30) # Michael Somos, Jun 04 2013

G.f.: Sum {k>0} k^5 * x^k / (1 - x^(2*k)).
From Amiram Eldar, Nov 01 2022: (Start)
Multiplicative with a(2^e) = 2^(5*e) and a(p^e) = (p^(5*e+5)-1)/(p^5-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^6, where c = 21*zeta(6)/128 = 0.166907... . (End)
Dirichlet g.f.: zeta(s)*zeta(s-5)*(1-1/2^s). - Amiram Eldar, Jan 08 2023

A002408 Expansion of 8-dimensional cusp form.

Original entry on oeis.org

0, 1, -8, 28, -64, 126, -224, 344, -512, 757, -1008, 1332, -1792, 2198, -2752, 3528, -4096, 4914, -6056, 6860, -8064, 9632, -10656, 12168, -14336, 15751, -17584, 20440, -22016, 24390, -28224, 29792, -32768, 37296, -39312, 43344, -48448, 50654, -54880, 61544, -64512, 68922

Offset: 0

N. J. A. Sloane and Mira Bernstein

Essentially the same as A007331.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
"For Gamma, it is known that any modular form is a weighted homogeneous polynomial in Theta_Z, which has weight 1/2, and the modular form delta_8(t) := e^(Pi i tau) Product_{m=1..oo} ((1 - e^(Pi i m tau)) (1 + e^(2 Pi i m tau)))^8 = e^(Pi i tau) - 8 e^(2 Pi i tau) + 28 e^(3 Pi i tau) - 64 e^(4 Pi i tau) + 126 e^(5 Pi i tau) ... of weight 4." [Elkies, p. 1242]

			G.f. = q - 8*q^2 + 28*q^3 - 64*q^4 + 126*q^5 - 224*q^6 + 344*q^7 ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 187.
Friedrich Hirzebruch, Thomas Berger, and Rainer Jung, Manifolds and Modular Forms, Vieweg 1994, p. 133.

T. D. Noe, Table of n, a(n) for n = 0..1000
Noam D. Elkies, Lattices, Linear Codes and Invariants, Part I, Notices of the Amer. Math. Soc., 47 (No. 10, Nov. 2000), 1238-1245, see p. 1242.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A000122, A000700, A007331, A010054, A045823, A121373, .

q*product((1-q^(2*k-1))^8*(1-q^(4*k))^8, k=1..75);

a[0] = 0; a[n_] := -(-1)^n*Sum[ Mod[n/d, 2]*d^3, {d, Divisors[n]}]; Table[a[n], {n, 0, 41}] (* Jean-François Alcover, Jan 27 2012, after Michael Somos *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^4] / QPochhammer[ q^2])^8, {q, 0, n}]; (* Michael Somos, May 25 2014 *)

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) / eta(x^2 + A))^8, n))}; /* Michael Somos, Jul 16 2004 */

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (prod(k=1, n, (1 -( k%4==0) * x^k) * (1 - (k%2==1) * x^k), 1 + A))^8, n))}; /* Michael Somos, Jul 16 2004 */

{a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, (n/d%2) * d^3))}; /* Michael Somos, May 31 2005 */

from sympy import divisors
def a(n): return 0 if n == 0 else -(-1)**n * sum([((n//d)%2) * d**3 for d in divisors(n)])
print([a(n) for n in range(101)])  # Indranil Ghosh, Jun 24 2017

A = ModularForms( Gamma0(4), 4, prec=70) . basis(); A[1] - 8*A[2] # _Michael Somos, May 25 2014

Expansion of (eta(q)* eta(q^4) / eta(q^2))^8 in powers of q. - Michael Somos, Jul 16 2004
Euler transform of period 4 sequence [-8, 0, -8, -8, ...]. - Michael Somos, Jul 16 2004
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = +u^4*w*v + 16*u^3*w*v^2 + 16*u^2*w^2*v^2 + 256*u^3*w^3 + 256*u^3*w^2*v + 4096*u^2*w^3*v + 4096*u*w^4*v + 4096*u*w^3*v^2 - u^2*v^4 - 16*u^2*w*v^3 - 256*u*w^2*v^3 - 256*w^2*v^4. - Michael Somos, May 31 2005
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^4*u6^4 + u1^3*u2*u3^3*u6 + 2*u1*u2^3*u3*u6^3 - u2^4*u3^4.
Expansion of q * psi(-q)^8 in powers of q where psi() is a Ramanujan theta function. - Michael Somos, Mar 20 2008
a(n) is multiplicative with a(2^e) = -8^e if e>0, a(p^e) = ((p^3)^(e+1) - 1) / (p^3 - 1). - Michael Somos, Mar 20 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 16 (t/i)^4 f(t) where q = exp(2 Pi i t).
G.f.: x * (Product_{k>0} (1 - x^(2*k-1)) * (1 - x^(4*k)))^8.
a(n) = -(-1)^n * A007331(n).
a(2*n) = -8 * A007331(n). a(2*n + 1) = A045823(n). - Michael Somos, May 25 2014
Dirichlet g.f.: zeta(s-3) * zeta(s) * (1 - 1/2^s) * (1 - 1/2^(s-4)). - Amiram Eldar, Nov 03 2023

A286180 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Product_{j>0} (1 + x^j) * (1 - x^(2*j)))^k in powers of x.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 1, 0, 1, 4, 3, 2, 0, 0, 1, 5, 6, 4, 2, 0, 0, 1, 6, 10, 8, 6, 0, 1, 0, 1, 7, 15, 15, 13, 3, 3, 0, 0, 1, 8, 21, 26, 25, 12, 6, 2, 0, 0, 1, 9, 28, 42, 45, 31, 14, 9, 0, 0, 0, 1, 10, 36, 64, 77, 66, 35, 24, 3, 2, 1, 0, 1, 11, 45

Offset: 0

Seiichi Manyama, May 07 2017

A(n, k) is the number of ways of writing n as the sum of k triangular numbers.

			Square array begins:
   1, 1, 1, 1,  1,  1, ...
   0, 1, 2, 3,  4,  5, ...
   0, 0, 1, 3,  6, 10, ...
   0, 1, 2, 4,  8, 15, ...
   0, 0, 2, 6, 13, 25, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-12 give A000007, A010054, A008441, A008443, A008438, A008439, A008440, A226252, A007331, A226253, A226254, A226255, A014787.

Main diagonal gives A106337.

Table[Function[k, SeriesCoefficient[Product[(1 + x^i) (1 - x^(2 i)), {i, Infinity}]^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten (* Michael De Vlieger, May 07 2017 *)

G.f. of column k: (Product_{j>0} (1 + x^j) * (1 - x^(2*j)))^k.

A340953 Number of ways to write n as an ordered sum of 8 nonzero triangular numbers.

Original entry on oeis.org

1, 0, 8, 0, 28, 8, 56, 56, 70, 176, 84, 336, 196, 448, 492, 504, 953, 616, 1456, 960, 1814, 1792, 1904, 3032, 2100, 4144, 3052, 4768, 4670, 5264, 6720, 5936, 8876, 7112, 10620, 9648, 11718, 12720, 13216, 15960, 15261, 19608, 17164, 23296, 21226, 25424, 26796, 27272, 32844

Offset: 8

Ilya Gutkovskiy, Jan 31 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..10000

Cf. A000217, A007331, A010054, A053603, A053604, A319818, A340949, A340950, A340951, A340952, A340954, A340955.

b:= proc(n, k) option remember; local r, t, d; r, t, d:= $0..2;
      if n=0 then `if`(k=0, 1, 0) else
      while t<=n do r:= r+b(n-t, k-1); t, d:= t+d, d+1 od; r fi
    end:
a:= n-> b(n, 8):
seq(a(n), n=8..56);  # Alois P. Heinz, Jan 31 2021

nmax = 56; CoefficientList[Series[(EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)) - 1)^8, {x, 0, nmax}], x] // Drop[#, 8] &

G.f.: (theta_2(sqrt(x)) / (2 * x^(1/8)) - 1)^8, where theta_2() is the Jacobi theta function.

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A014787 Expansion of Jacobi theta constant (theta_2/2)^12.

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A014809 Expansion of Jacobi theta constant (theta_2/2)^24.

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A206622 G.f.: Product_{n>0} ( (1+x^n)/(1-x^n) )^(n^2).

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A076577 Sum of squares of divisors d of n such that n/d is odd.

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A226255 Number of ways of writing n as the sum of 11 triangular numbers.

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A035016 Fourier coefficients of E_{0,4}.

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A096960 a(n) = Sum_{0

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