A096960 -id:A096960 - cp's OEIS frontend

A007331 Fourier coefficients of E_{infinity,4}.

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

Offset: 0

N. J. A. Sloane, Mira Bernstein

E_{infinity,4} is the unique normalized weight-4 modular form for Gamma_0(2) with simple zeros at i*infinity. Since this has level 2, it is not a cusp form, in contrast to A002408.
a(n+1) is the number of representations of n as a sum of 8 triangular numbers (from A000217). See the Ono et al. link, Theorem 5.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
a(n) gives the sum of cubes of divisors d of n such that n/d is odd. This is called sigma^#3(n) in the Ono et al. link. See a formula below. - _Wolfdieter Lang, Jan 12 2017

			G.f. = q + 8*q^2 + 28*q^3 + 64*q^4 + 126*q^5 + 224*q^6 + 344*q^7 + 512*q^8 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 139, Ex (ii).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1001 from T. D. Noe)
B. Brent, Quadratic Minima and Modular Forms, Experimental Mathematics, v.7 no.3, 257-274.
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. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 187.
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).
Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
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. Theorem 5.
H. Rosengren, Sums of triangular numbers from the Frobenius determinant, arXiv:math/0504272 [math.NT], 2005.
Index entries for sequences mentioned by Glaisher.

Cf. A002408, A004017, A035016, A045825, A076577, A096960, A222072.

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, A076577.

Basis( ModularForms( Gamma0(2), 4), 10) [2]; /* Michael Somos, May 27 2014 */

nmax:=40: seq(coeff(series(x*(product((1-x^k)^8*(1+x^k)^16, k=1..nmax)), x, n+1), x, n), n=0..nmax); # Vaclav Kotesovec, Oct 14 2015

Prepend[Table[Plus @@ (Select[Divisors[k + 1], OddQ[(k + 1)/#] &]^3), {k, 0, 39}], 0] (* Ant King, Dec 04 2010 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(1/2)]^8 / 256, {q, 0, n}]; (* Michael Somos, Jun 04 2013 *)
a[ n_] := If[ n < 1, 0, Sum[ d^3 Boole[ OddQ[ n/d]], {d, Divisors[ n]}]]; (* Michael Somos, Jun 04 2013 *)
f[n_] := Total[(2n/Select[ Divisors[ 2n], Mod[#, 4] == 2 &])^3]; Flatten[{0, Array[f, 40] }] (* Robert G. Wilson v, Mar 26 2015 *)
nmax=60; CoefficientList[Series[x*Product[(1-x^k)^8 * (1+x^k)^16, {k,1,nmax}],{x,0,nmax}], x] (* Vaclav Kotesovec, Oct 14 2015 *)
QP = QPochhammer; s = q * (QP[-1, q]/2)^16 * QP[q]^8 + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Dec 01 2015, adapted from PARI *)

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

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / eta(x + A))^8, n))}; /* Michael Somos, May 31 2005 */

a(n)=my(e=valuation(n,2)); 8^e * sigma(n/2^e, 3) \\ Charles R Greathouse IV, Sep 09 2014

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

ModularForms( Gamma0(2), 4, prec=33).1; # Michael Somos, Jun 04 2013

G.f.: q * Product_{k>=1} (1-q^k)^8 * (1+q^k)^16. - corrected by Vaclav Kotesovec, Oct 14 2015

a(n) = Sum_{0

G.f.: Sum_{n>0} n^3*x^n/(1-x^(2*n)). - Vladeta Jovovic, Oct 24 2002

Expansion of Jacobi theta constant theta_2(q)^8 / 256 in powers of q.

Expansion of eta(q^2)^16 / eta(q)^8 in powers of q. - Michael Somos, May 31 2005

Expansion of x * psi(x)^8 in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Jan 15 2012

Expansion of (Q(x) - Q(x^2)) / 240 in powers of x where Q() is a Ramanujan Lambert series. - Michael Somos, Jan 15 2012

Expansion of E_{gamma,2}^2 * E_{0,4} in powers of q.

Euler transform of period 2 sequence [8, -8, ...]. - Michael Somos, May 31 2005

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^3 - u^2*w + 16*u*v*w - 32*v^2*w + 256*v*w^2. - Michael Somos, May 31 2005

G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = 16^(-1) (t / i)^4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A035016. - Michael Somos, Jan 11 2009

Multiplicative with a(2^e) = 2^(3e), a(p^e) = (p^(3(e+1))-1)/(p^3-1). - Mitch Harris, Jun 13 2005

Dirichlet convolution of A154955 by A001158. Dirichlet g.f. zeta(s)*zeta(s-3)*(1-1/2^s). - R. J. Mathar, Mar 31 2011

A002408(n) = -(-1)^n * a(n).

Convolution square of A008438. - Michael Somos, Jun 15 2014

a(1) = 1, a(n) = (8/(n-1))*Sum_{k=1..n-1} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, May 06 2017

Sum_{k=1..n} a(k) ~ c * n^4, where c = Pi^4/384 = 0.253669... (A222072). - Amiram Eldar, Oct 19 2022

Additional comments from Barry Brent (barryb(AT)primenet.com)
Wrong Maple program replaced by Vaclav Kotesovec, Oct 14 2015
a(0)=0 prepended by Vaclav Kotesovec, Oct 14 2015

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

A206624 G.f.: Product_{n>0} ( (1+x^n)/(1-x^n) )^(n^4).

Original entry on oeis.org

1, 2, 34, 228, 1414, 8872, 52876, 301136, 1662614, 8929406, 46738920, 239036116, 1197187780, 5882369976, 28397283056, 134864166352, 630819797174, 2908948327780, 13236421303742, 59477002686404, 264104800719672, 1159649708139680, 5037895127964316

Offset: 0

Paul D. Hanna, Feb 12 2012

nonn

Convolution of A023873 and A248883. - Vaclav Kotesovec, Aug 19 2015
In general, for m >= 0, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^(k^m), then a(n) ~ ((2^(m+2)-1) * Gamma(m+2) * Zeta(m+2) / (2^(2*m+3) * n))^((1-2*Zeta(-m))/(2*m+4)) * exp((m+2)/(m+1) * ((2^(m+2)-1) * n^(m+1) * Gamma(m+2) * Zeta(m+2) / 2^(m+1))^(1/(m+2)) + Zeta'(-m)) / sqrt((m+2)*Pi*n). - Vaclav Kotesovec, Aug 19 2015
If m is even and m >= 2, then can be simplified as: a(n) ~ ((2^(m+2)-1) * Gamma(m+2) * Zeta(m+2) / (2^(2*m+3) * n))^(1/(2*m+4)) * exp((m+2)/(m+1) * ((2^(m+2)-1) * n^(m+1) * Gamma(m+2) * Zeta(m+2) / 2^(m+1))^(1/(m+2)) + (-1)^(m/2) * Gamma(m+1) * Zeta(m+1) / (2^(m+1) * Pi^m)) / sqrt((m+2)*Pi*n). - Vaclav Kotesovec, Aug 19 2015

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 88*x^3 + 398*x^4 + 1768*x^5 + 7508*x^6 +...
where A(x) = (1+x)/(1-x) * (1+x^2)^16/(1-x^2)^16 * (1+x^3)^81/(1-x^3)^81 *...
Also, A(x) = Euler transform of [2,31,162,496,1250,2511,4802,7936,...]:
A(x) = 1/((1-x)^2*(1-x^2)^31*(1-x^3)^162*(1-x^4)^496*(1-x^5)^1250*(1-x^6)^2511*...).

Seiichi Manyama, Table of n, a(n) for n = 0..2916 (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. A015128 (m=0), A156616 (m=1), A206622 (m=2), A206623 (m=3), A001160 (sigma_5).

nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^(k^4), {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^4)),n)}

{a(n)=polcoeff(exp(sum(m=1, n, (sigma(2*m, 5)-sigma(m, 5))/16*x^m/m)+x*O(x^n)), n)}

{a(n)=local(InvEulerGF=x*(2+31*x+152*x^2+341*x^3+460*x^4+341*x^5+152*x^6+31*x^7+2*x^8)/(1-x^2+x*O(x^n))^5); polcoeff(1/prod(k=1,n,(1-x^k+x*O(x^n))^polcoeff(InvEulerGF,k)),n)}
for(n=0,30,print1(a(n),", "))

G.f.: exp( Sum_{n>=1} (sigma_5(2*n) - sigma_5(n))/16 * x^n/n ), where sigma_5(n) is the sum of 5th powers of divisors of n (A001160).
Inverse Euler transform has g.f.: x*(2 + 31*x + 152*x^2 + 341*x^3 + 460*x^4 + 341*x^5 + 152*x^6 + 31*x^7 + 2*x^8)/(1-x^2)^5.
a(n) ~ exp(3*2^(2/3)*Pi*n^(5/6)/5 + 3*Zeta(5)/(4*Pi^4)) / (2^(7/6) * 3^(1/2) * n^(7/12)), where Zeta(5) = A013663. - Vaclav Kotesovec, Aug 19 2015
a(0) = 1, a(n) = (2/n)*Sum_{k=1..n} A096960(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 30 2017

A096963 a(n) = Sum {0

Original entry on oeis.org

1, 2048, 177148, 4194304, 48828126, 362799104, 1977326744, 8589934592, 31381236757, 100000002048, 285311670612, 743012564992, 1792160394038, 4049565171712, 8649804864648, 17592186044416, 34271896307634

Offset: 1

Ralf Stephan, Jul 18 2004

This is the member k=11 of the k-family sigma^#_k(n) := Sum {0

This notation appears in the Ono et al. link, Theorem 5 (with k=3, see A007331) and Theorem 8 (with k=11). - Wolfdieter Lang, Jan 13 2017

			G.f. = q + 2048*q^2 + 177148*q^3 + 4194304*q^4 + 48828126*q^5 + ...

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).
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, Case 24, Theorem 8.
Index entries for sequences mentioned by Glaisher.

Cf. A007331, A002131, A013670, A096960, A096961, A096962.

A := Basis( ModularForms( Gamma0(2), 12), 18); A[2] + 2048*A[3] + 177148*A[4]; /* Michael Somos, Nov 30 2014 */

a[ n_] := If[ n < 1, 0, Sum[ d^11 Boole[ OddQ[ n/d]], {d, Divisors[ n]}]]; (* Michael Somos, Nov 30 2014 *)
a[ n_] := SeriesCoefficient[ With[{u1 = QPochhammer[ q]^8, u4 = QPochhammer[ q^4]^8}, q (u1^4 + 2072 q u4 u1^3 + 210048 q^2 u4^2 u1^2 + 5660672 q^3 u4^3 u1 + 45285376 q^4 u4^4) / u1 ], {q, 0, n}]; (* Michael Somos, Nov 30 2014 *)

{a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d^11))}; /* Michael Somos, Nov 30 2014 */

ModularForms( Gamma0(2), 12, prec=18).3; # Michael Somos, Nov 30 2014

G.f.: Sum_{n>0} n^11 * x^n / (1 - x^(2*n)).

a(n) = Sum {0

From Amiram Eldar, Nov 02 2022: (Start)

Multiplicative with a(2^e) = 2^(11*e) and a(p^e) = (p^(11*e+11)-1)/(p^11-1) for p > 2.

Sum_{k=1..n} a(k) ~ c * n^12, where c = 1365*zeta(12)/16384 = 691*Pi^12/7664025600 = 0.0833334904... . (End)

Dirichlet g.f.: zeta(s)*zeta(s-11)*(1-1/2^s). - Amiram Eldar, Jan 09 2023

A096961 a(n) = Sum_{0

Original entry on oeis.org

1, 128, 2188, 16384, 78126, 280064, 823544, 2097152, 4785157, 10000128, 19487172, 35848192, 62748518, 105413632, 170939688, 268435456, 410338674, 612500096, 893871740, 1280016384, 1801914272, 2494358016, 3404825448

Offset: 1

Ralf Stephan, Jul 18 2004

			G.f. = q + 128*q^2 + 2188*q^3 + 16384*q^4 + 78126*q^5 + 280064*q^6 + 823544*q^7 + ...

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, A008410, A013666, A096960, A096962, A096963.

A := Basis( ModularForms( Gamma0(2), 8), 24); A[2] + 128*A[3]; /* Michael Somos, Nov 30 2014 */

a[ n_] := SeriesCoefficient[ With[{u1 = QPochhammer[ q]^8, u2 = QPochhammer[ q^2]^8, u4 = QPochhammer[ q^4]^8}, q u2 (u1^2 + 136 q u4 u1 + 2176 q^2 u4^2 ) / u1], {q, 0, n}]; (* Michael Somos, Jun 04 2013 *)
a[ n_] := If[ n < 1, 0, Sum[ d^7 Mod[ n/d, 2], {d, Divisors[ n]}]]; (* Michael Somos, Jan 09 2015 *)

{a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d^7))};

ModularForms( Gamma0(2), 8, prec=24).2; # Michael Somos, Jun 04 2013

G.f.: Sum_{k>0} k^7 * x^k / (1 - x^(2*k)).
Expansion of (E_8(q) - E_8(q^2)) / 480 in powers of q where E_8() is an Eisenstein series (A008410). - Michael Somos, Jan 09 2015
From Amiram Eldar, Nov 02 2022: (Start)
Multiplicative with a(2^e) = 2^(7*e) and a(p^e) = (p^(7*e+7)-1)/(p^7-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^8, where c = 255*zeta(8)/2048 = 17*Pi^8/1290240 = 0.125019... . (End)
Dirichlet g.f.: zeta(s)*zeta(s-7)*(1-1/2^s). - Amiram Eldar, Jan 09 2023

A096962 a(n) = Sum_{0

Original entry on oeis.org

1, 512, 19684, 262144, 1953126, 10078208, 40353608, 134217728, 387440173, 1000000512, 2357947692, 5160042496, 10604499374, 20661047296, 38445332184, 68719476736, 118587876498, 198369368576, 322687697780, 512000262144

Offset: 1

Ralf Stephan, Jul 18 2004

			G.f. = q + 512*q^2 + 19684*q^3 + 262144*q^4 + 1953126*q^5 + 10078208*q^6 + ...

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.

Cf. A007331, A013668, A096960, A096961, A096963.

A := Basis( ModularForms( Gamma0(2), 10), 21); A[2] + 512*A[3]; /* Michael Somos, Aug 25 2014 */

a[ n_] := If[ n < 1, 0, Sum[ d^9 Boole[ OddQ[ n/d]], {d, Divisors[ n]}]]; (* Michael Somos, Jun 04 2013 *)
a[ n_] := SeriesCoefficient[ With[{u1 = QPochhammer[ q]^8, u2 = QPochhammer[ q^2]^4, u4 = QPochhammer[ q^4]^8}, q u2 (u1 + 32 q u4) (u1^2 + 496 q u4 u1 + 7936 q^2 u4^2 ) / u1], {q, 0, n}]; (* Michael Somos, Jun 04 2013 *)

{a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d^9))}; /* Michael Somos, Jun 04 2013 */

{a(n) = local(A, A1, A2, A4); if( n<1, 0, n--; A = x * O(x^n); A1 = eta(x + A)^8; A2 = eta(x^2 + A)^4; A4 = eta(x^4 + A)^8; polcoeff( A2 * (A1 + 32*x * A4) * (A1^2 + 496*x * A1*A4 + 7936*x^2 * A4^2) / A1, n))}; /* Michael Somos, Jun 04 2013 */

ModularForms( Gamma0(2), 10, prec=33).2; # Michael Somos, Jun 04 2013

G.f.: Sum_{k>0} k^9 * x^k / (1 - x^(2*k)).
From Amiram Eldar, Nov 02 2022: (Start)
Multiplicative with a(2^e) = 2^(9*e) and a(p^e) = (p^(9*e+9)-1)/(p^9-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^10, where c = 1023*zeta(10)/10240 = 31*Pi^10/29030400 = 0.100001704136... . (End)
Dirichlet g.f.: zeta(s)*zeta(s-9)*(1-1/2^s). - Amiram Eldar, Jan 09 2023

A285989 a(0) = 0, a(n) = Sum_{0 0.

Original entry on oeis.org

0, 1, 16, 82, 256, 626, 1312, 2402, 4096, 6643, 10016, 14642, 20992, 28562, 38432, 51332, 65536, 83522, 106288, 130322, 160256, 196964, 234272, 279842, 335872, 391251, 456992, 538084, 614912, 707282, 821312, 923522, 1048576, 1200644, 1336352, 1503652, 1700608

Offset: 0

Seiichi Manyama, Apr 30 2017

Multiplicative because this sequence is the Dirichlet convolution of A000035 and A000583 which are both multiplicative. - Andrew Howroyd, Aug 05 2018

Robert Israel, Table of n, a(n) for n = 0..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.

Sum_{0A002131 (k=1), A076577 (k=2), A007331 (k=3), this sequence (k=4), A096960 (k=5), A096961 (k=7), A096962 (k=9), A096963 (k=11).

Cf. A000035, A000583, A007814, A013663, A051001, A206623, A285990.

f:= n -> add((n/d)^4, d = numtheory:-divisors(n/2^padic:-ordp(n,2))); # Robert Israel, Apr 30 2017

{0}~Join~Table[DivisorSum[n, Mod[#, 2] (n/#)^4 &], {n, 36}] (* Michael De Vlieger, Aug 05 2018 *)

a(n)={sumdiv(n, d, (d%2)*(n/d)^4)} \\ Andrew Howroyd, Aug 05 2018

a(n) = A051001(n)*16^A007814(n) for n >= 1. - Robert Israel, Apr 30 2017
From Amiram Eldar, Nov 01 2022: (Start)
Multiplicative with a(2^e) = 2^(4*e) and a(p^e) = (p^(4*e+4)-1)/(p^4-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^5, where c = 31*zeta(5)/160 = 0.200904... . (End)
Dirichlet g.f.: zeta(s)*zeta(s-4)*(1-1/2^s). - Amiram Eldar, Jan 08 2023

A322082 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n, n/d odd} d^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 4, 4, 1, 1, 8, 10, 4, 2, 1, 16, 28, 16, 6, 2, 1, 32, 82, 64, 26, 8, 2, 1, 64, 244, 256, 126, 40, 8, 1, 1, 128, 730, 1024, 626, 224, 50, 8, 3, 1, 256, 2188, 4096, 3126, 1312, 344, 64, 13, 2, 1, 512, 6562, 16384, 15626, 7808, 2402, 512, 91, 12, 2, 1, 1024, 19684, 65536, 78126, 46720, 16808, 4096, 757, 104, 12, 2

Offset: 1

Ilya Gutkovskiy, Nov 26 2018

			Square array begins:
  1,  1,   1,    1,     1,     1,  ...
  1,  2,   4,    8,    16,    32,  ...
  2,  4,  10,   28,    82,   244,  ...
  1,  4,  16,   64,   256,  1024,  ...
  2,  6,  26,  126,   626,  3126,  ...
  2,  8,  40,  224,  1312,  7808,  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)
Index entries for sequences mentioned by Glaisher

Columns k=0..12 give A001227, A002131, A076577, A007331, A285989, A096960, A321817, A096961, A321818, A096962, A321819, A096963, A321820.

Cf. A109974, A279394, A279396, A285425, A322081, A322083, A322084.

Table[Function[k, Sum[Boole[OddQ[n/d]] d^k, {d, Divisors[n]}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[j^k x^j/(1 - x^(2 j)), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten

T(n,k)={sumdiv(n, d, if(n/d%2, d^k))}
for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 26 2018

G.f. of column k: Sum_{j>=1} j^k*x^j/(1 - x^(2*j)).

A285991 Expansion of Product_{n>0} ((1-x^n)/(1+x^n))^(n^4) in powers of x.

Original entry on oeis.org

1, -2, -30, -100, 262, 3672, 13836, -80, -264810, -1421438, -3019032, 7630764, 89648580, 358974280, 548677872, -2390377936, -20531491146, -74635378020, -110275527170, 425036176572, 3669041188152, 13597190512480, 23995331740700, -45340748171760

Offset: 0

Seiichi Manyama, Apr 30 2017

sign

Product_{n>0} ((1-x^n)/(1+x^n))^(n^m): A002448 (m=0), A285675 (m=1), A285988 (m=2), A285990 (m=3), this sequence (m=4).

Cf. A096960, A206624.

a(0) = 1, a(n) = -(2/n)*Sum_{k=1..n} A096960(k)*a(n-k) for n > 0.

G.f.: exp(Sum_{k>=1} (sigma_5(k) - sigma_5(2*k))*x^k/(16*k)). - Ilya Gutkovskiy, Apr 14 2019

Showing 1-9 of 9 results.

cp's OEIS Frontend

A007331 Fourier coefficients of E_{infinity,4}.

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

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A206624 G.f.: Product_{n>0} ( (1+x^n)/(1-x^n) )^(n^4).

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A096963 a(n) = Sum {0

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A096961 a(n) = Sum_{0

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A096962 a(n) = Sum_{0

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