A050468 -id:A050468 - cp's OEIS frontend

A321835 a(n) = Sum_{d|n, n/d==1 mod 4} d^11 - Sum_{d|n, n/d==3 mod 4} d^11.

1, 2048, 177146, 4194304, 48828126, 362795008, 1977326742, 8589934592, 31380882463, 100000002048, 285311670610, 743004176384, 1792160394038, 4049565167616, 8649707208396, 17592186044416, 34271896307634, 64268047284224, 116490258898218

Offset: 1

N. J. A. Sloane, Nov 24 2018

Seiichi Manyama, 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. A101455.
Cf. A321807 - A321836 for related sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, A321832, A321833, A321834, this sequence, A321836.

s[n_,r_] := DivisorSum[n, #^11 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(11*e+11) - s[p]^(e+1))/(p^11 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)

apply( a(n)=sumdiv(n,d,if(bittest(n\d,0),(2-n\d%4)*d^11)), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^11*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
From Amiram Eldar, Nov 04 2023: (Start)
Multiplicative with a(p^e) = (p^(11*e+11) - A101455(p)^(e+1))/(p^11 - A101455(p)).
Sum_{k=1..n} a(k) ~ c * n^12 / 12, where c = beta(12) = 0.99999812235..., and beta is the Dirichlet beta function. (End)
a(n) = Sum_{d|n} (n/d)^11*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A000144 Number of ways of writing n as a sum of 10 squares.

Original entry on oeis.org

1, 20, 180, 960, 3380, 8424, 16320, 28800, 52020, 88660, 129064, 175680, 262080, 386920, 489600, 600960, 840500, 1137960, 1330420, 1563840, 2050344, 2611200, 2986560, 3358080, 4194240, 5318268, 5878440, 6299520, 7862400, 9619560

Offset: 0

N. J. A. Sloane

			G.f. = 1 + 20*x + 180*x^2 + 960*x^3 + 3380*x^4 + 8424*x^5 + 16320*x^6 + ...

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 314.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Chelsea Publishing Company, New York 1959, p. 135 section 9.3. MR0106147 (21 #4881)

T. D. Noe, Table of n, a(n) for n = 0..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.
Shi-Chao Chen, Congruences for rs(n), Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032.
J. Liouville, Nombre des représentations d’un entier quelconque sous la forme d’une somme de dix carrés, Journal de mathématiques pures et appliquées 2e série, tome 11 (1866), p. 1-8.
Index entries for sequences related to sums of squares

Row d=10 of A122141 and of A319574, 10th column of A286815.

(sum(x^(m^2),m=-10..10))^10;
# Alternative:
A000144list := proc(len) series(JacobiTheta3(0, x)^10, x, len+1);
seq(coeff(%, x, j), j=0..len-1) end: A000144list(30); # Peter Luschny, Oct 02 2018

Table[SquaresR[10, n], {n, 0, 30}] (* Ray Chandler, Jun 29 2008; updated by T. D. Noe, Jan 23 2012 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^10, {q, 0, n}]; (* Michael Somos, Aug 26 2015 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^k)^10 * (1 + x^k)^30 / (1 + x^(2*k))^20, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 24 2017 *)

{a(n) = if( n<0, 0, polcoeff( sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n))^10, n))}; /* Michael Somos, Sep 12 2005 */

Q = DiagonalQuadraticForm(ZZ, [1]*10)
Q.representation_number_list(37) # Peter Luschny, Jun 20 2014

Euler transform of period 4 sequence [ 20, -30, 20, -10, ...]. - Michael Somos, Sep 12 2005
Expansion of eta(q^2)^50 / (eta(q) * eta(q^4))^20 in powers of q. - Michael Somos, Sep 12 2005
a(n) = 4/5 * (A050456(n) + 16*A050468(n) + 8*A030212(n)) if n>0. - Michael Somos, Sep 12 2005
a(n) = (20/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017

Extended by Ray Chandler, Nov 28 2006

A050463 a(n) = Sum_{d|n, n/d=1 mod 4} d^4.

Original entry on oeis.org

1, 16, 81, 256, 626, 1296, 2401, 4096, 6562, 10016, 14641, 20736, 28562, 38416, 50706, 65536, 83522, 104992, 130321, 160256, 194482, 234256, 279841, 331776, 391251, 456992, 531522, 614656, 707282, 811296, 923521, 1048576, 1185922

Offset: 1

N. J. A. Sloane, Dec 23 1999

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A050448, A050467, A050468, A013663, A285989.

Cf. A050460, A050461, A050462.

a[n_] := DivisorSum[n, #^4 &, Mod[n/#, 4] == 1 &]; Array[a, 50] (* Amiram Eldar, Jul 08 2023 *)

a(n) = sumdiv(n, d, (n/d % 4 == 1) * d^4); \\ Amiram Eldar, Nov 05 2023

From Amiram Eldar, Nov 05 2023: (Start)
a(n) = A285989(n) - A050467(n).
a(n) = A050468(n) + A050467(n).
a(n) = (A050468(n) + A285989(n))/2.
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = 5*Pi^5/3072 + 31*zeta(5)/64 = 1.000340795436113... . (End)

Offset changed from 0 to 1 by Seiichi Manyama, Jul 08 2023

A050467 a(n) = Sum_{d|n, n/d=3 mod 4} d^4.

Original entry on oeis.org

0, 0, 1, 0, 0, 16, 1, 0, 81, 0, 1, 256, 0, 16, 626, 0, 0, 1296, 1, 0, 2482, 16, 1, 4096, 0, 0, 6562, 256, 0, 10016, 1, 0, 14722, 0, 626, 20736, 0, 16, 28562, 0, 0, 39712, 1, 256, 50706, 16, 1, 65536, 2401, 0, 83522, 0, 0, 104992, 626, 4096, 130402

Offset: 1

N. J. A. Sloane, Dec 23 1999

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A013663, A050463, A050468, A285989.

Cf. A050464, A050465, A050466.

Table[Total[Select[Divisors[n],Mod[n/#,4]==3&]^4],{n,60}] (* Harvey P. Dale, Jun 10 2023 *)
a[n_] := DivisorSum[n, #^4 &, Mod[n/#, 4] == 3 &]; Array[a, 50] (* Amiram Eldar, Nov 05 2023 *)

a(n) = sumdiv(n, d, (n/d % 4 == 3) * d^4); \\ Amiram Eldar, Nov 05 2023

From Amiram Eldar, Nov 05 2023: (Start)
a(n) = A285989(n) - A050463(n).
a(n) = A050463(n) - A050468(n).
a(n) = (A285989(n) - A050468(n))/2.
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = 31*zeta(5)/64 - 5*Pi^5/3072 = 0.00418296735902... . (End)

Offset corrected by Amiram Eldar, Nov 05 2023

A322084 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n, n/d==1 (mod 4)} d^k - Sum_{d|n, n/d==3 (mod 4)} d^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 4, 2, 1, 1, 8, 8, 4, 2, 1, 16, 26, 16, 6, 0, 1, 32, 80, 64, 26, 4, 0, 1, 64, 242, 256, 126, 32, 6, 1, 1, 128, 728, 1024, 626, 208, 48, 8, 1, 1, 256, 2186, 4096, 3126, 1280, 342, 64, 7, 2, 1, 512, 6560, 16384, 15626, 7744, 2400, 512, 73, 12, 0, 1, 1024, 19682, 65536, 78126, 46592, 16806, 4096, 703, 104, 10, 0

Offset: 1

Ilya Gutkovskiy, Nov 26 2018

			Square array begins:
  1,  1,   1,    1,     1,     1,  ...
  1,  2,   4,    8,    16,    32,  ...
  0,  2,   8,   26,    80,   242,  ...
  1,  4,  16,   64,   256,  1024,  ...
  2,  6,  26,  126,   626,  3126,  ...
  0,  4,  32,  208,  1280,  7744,  ...

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 A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, A321832, A321833, A321834, A321835, A321836.

Cf. A109974, A279394, A279396, A285425, A322081, A322082, A322083.

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(d%2, (-1)^((d-1)/2)*(n/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)).

A204342 a(n) = (-1)^n * Sum_{2m + 1 | 2n + 1} (-1)^m (2*m + 1)^4.

Original entry on oeis.org

1, 80, 626, 2400, 6481, 14640, 28562, 50080, 83522, 130320, 192000, 279840, 391251, 524960, 707282, 923520, 1171200, 1502400, 1874162, 2284960, 2825762, 3418800, 4057106, 4879680, 5762401, 6681760, 7890482, 9164640, 10425600

Offset: 0

Michael Somos, Jan 14 2012

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

			1 + 80*x + 626*x^2 + 2400*x^3 + 6481*x^4 + 14640*x^5 + 28562*x^6 + ...
q + 80*q^3 + 626*q^5 + 2400*q^7 + 6481*q^9 + 14640*q^11 + 28562*q^13 + ...
a(1) = 80 since (-1)^1 * ( (-1)^0 * 1^4 + (-1)^1 * 3^4) = 80 where 1 and 3 are the odd divisors of 3 = 2*1 + 1.

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. 2, p. 315.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A050468.

Cf. A000122, A000700, A010054, A121373.

QP:= QPochhammer[q]; a[n_]:= SeriesCoefficient[QP[q^2]^14* (QP[q]^8 + 80*q*QP[q^4]^8)/(QP[q]^8*QP[q^4]^4), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Apr 11 2018 *)

{a(n) = if( n<0, 0, (-1)^n * sumdiv( 2*n + 1, d, (-1)^(d\2) *  d^4))}

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^14 * (eta(x + A)^8 + 80 * x * eta(x^4 + A)^8) / (eta(x + A)^8 * eta(x^4 + A)^4), n))}

Expansion of phi(x)^4 * psi(x^2)^2 * (phi(x)^4 + 64 * x * psi(x^2)^4) in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of q^(-1/2) * eta(q^2)^14 * (eta(q)^8 + 80 * q * eta(q^4)^8) / (eta(q)^8 * eta(q^4)^4) in powers of q.
a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = ((p^4)^(e+1) + 1) / (p^4 + 1) if p == 3 (mod 4), b(p^e) = ((p^4)^(e+1) - 1) / (p^4 - 1) if p == 1 (mod 4).
G.f.: Sum_{k > 0} (2*k - 1)^4 * x^(2*k - 1) / (1 + x^(4*k - 2)).
a(n) = A050468(2*n + 1).
Sum_{k=1..n} a(k) ~ c * n^5, where c = Pi^5/96 = 3.187705... . - Amiram Eldar, Dec 29 2023

A204372 Expansion of phi(x)^2 * (5 * phi(-x)^8 + 64 * x * psi(-x)^8) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

5, 4, 4, -320, 4, 2504, -320, -9600, 4, 25924, 2504, -58560, -320, 114248, -9600, -200320, 4, 334088, 25924, -521280, 2504, 768000, -58560, -1119360, -320, 1565004, 114248, -2099840, -9600, 2829128, -200320, -3694080, 4, 4684800

Offset: 0

Michael Somos, Jan 14 2012

sign

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

			G.f. = 5 + 4*x + 4*x^2 - 320*x^3 + 4*x^4 + 2504*x^5 - 320*x^6 - 9600*x^7 + 4*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A050456, A050468.

A := Basis( ModularForms( Gamma1(4), 5), 34); 5*A[1] + 4*A[2] + 4*A[3]; /* Michael Somos, May 04 2015 */

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^2 (5 EllipticTheta[ 4, 0, q]^8 + 4 EllipticTheta[ 2, Pi/4, q^(1/2)]^8), {q, 0, n}]; (* Michael Somos, May 03 2015 *)
a[ n_] := If[ n < 1, 5 Boole[n == 0], 4 DivisorSum[ n, #^4 KroneckerSymbol[ -4, #] &]]; (* Michael Somos, May 04 2015 *)

{a(n) = if( n<1, 5 * (n==0), 4 * sumdiv( n, d, d^4 * kronecker( -4, d)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^2 + A)^2 * (5 * eta(x + A)^8 / eta(x^4 + A)^4 + 64 * x * eta(x^4 + A)^4 ), n))};

Expansion of eta(q)^4 * eta(q^2)^2 * (5 * eta(q)^8 / eta(q^4)^4 + 64 * q * eta(q^4)^4 ) in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (4*t)) = 2048 (t/i)^5 g(t) where q = exp(2*Pi*i*t) and g(t) is the g.f. for A050468.
G.f.: 5 + 4 * Sum_{k>0} (-1)^(k-1) * (2*k - 1)^4 * x^(2*k - 1) / (1 - x^(2*k - 1)).
a(n) = 4 * A050456(n) if n>0.

A204386 Expansion of (theta_2(q)^8 + 4 * theta_2(q^2)^8) / 256 in powers of q^2.

Original entry on oeis.org

1, 12, 28, 96, 126, 336, 344, 768, 757, 1512, 1332, 2688, 2198, 4128, 3528, 6144, 4914, 9084, 6860, 12096, 9632, 15984, 12168, 21504, 15751, 26376, 20440, 33024, 24390, 42336, 29792, 49152, 37296, 58968, 43344, 72672, 50654, 82320, 61544, 96768

Offset: 1

Michael Somos, Jan 15 2012

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

			x + 12*x^2 + 28*x^3 + 96*x^4 + 126*x^5 + 336*x^6 + 344*x^7 + 768*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A004018, A007331, A045823, A050468.

Cf. A000122, A000700, A010054, A121373.

a[n_]:= SeriesCoefficient[(EllipticTheta[2, 0, q^(1/2)]^8 + 4*EllipticTheta[2, 0, q]^8)/256, {q, 0, n}];  Table[a[n], {n,1,50}] (* G. C. Greubel, Apr 13 2018 *)
CoefficientList[Series[(EllipticTheta[2,0,q^(1/2)]^8 +4*EllipticTheta[2, 0, q]^8)/ 256, {q, 0, 50}], q] (* Vaclav Kotesovec, Apr 13 2018 *)

{a(n) = if( n<1, 0, if( n%2, sigma( n, 3), 12 * sumdiv( n/2, d, (n/2/d%2) * d^3)))}

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

Expansion of x * psi(x)^8 + 4 * x^2 * psi(x^2)^8 in powers of x where psi() is a Ramanujan theta function.
Expansion of (eta(q^2)^2 / eta(q))^8 + 4 * (eta(q^4)^2 / eta(q^2))^8 in powers of q.
a(n) is multiplicative with a(2^e) = 3/2 * 8^e if e>0, a(p^e) = ((p^3) ^ (e+1) - 1) / (p^3 - 1).
a(2*n + 1) = A045823(n). a(2*n) = 12 * A007331(n).
Convolution of this sequence with A004018 is A050468.
From Amiram Eldar, Sep 12 2023: (Start)
Dirichlet g.f.: (1 + 1/2^(s-2)) * (1 - 1/2^s) * zeta(s-3) * zeta(s).
Sum_{k=1..n} a(k) ~ c * n^4, where c = 5*Pi^4/1536 = 0.317086... . (End)

cp's OEIS Frontend

A321835 a(n) = Sum_{d|n, n/d==1 mod 4} d^11 - Sum_{d|n, n/d==3 mod 4} d^11.

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A000144 Number of ways of writing n as a sum of 10 squares.

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A050463 a(n) = Sum_{d|n, n/d=1 mod 4} d^4.

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A050467 a(n) = Sum_{d|n, n/d=3 mod 4} d^4.

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A322084 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n, n/d==1 (mod 4)} d^k - Sum_{d|n, n/d==3 (mod 4)} d^k.

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A204342 a(n) = (-1)^n * Sum_{2*m + 1 | 2*n + 1} (-1)^m (2*m + 1)^4.

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A204372 Expansion of phi(x)^2 * (5 * phi(-x)^8 + 64 * x * psi(-x)^8) in powers of x where phi(), psi() are Ramanujan theta functions.

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A204342 a(n) = (-1)^n * Sum_{2m + 1 | 2n + 1} (-1)^m (2*m + 1)^4.