A033715 -id:A033715 - cp's OEIS frontend

A138811 Theta series of quadratic form x^2 + xy + 11y^2.

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

Offset: 0

Michael Somos, Mar 31 2008, Apr 05 2008

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

			G.f. = 1 + 2*q + 2*q^4 + 2*q^9 + 4*q^11 + 4*q^13 + 2*q^16 + 4*q^17 + 4*q^23 + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A035147.
Cf. A000122, A000700, A010054, A121373.
Number of integer solutions to f(x,y) = n where f(x,y) is the principal binary quadratic form with discriminant d: A004016 (d=-3), A004018 (d=-4), A002652 (d=-7), A033715 (d=-8), A028609 (d=-11), A028641 (d=-19), this sequence (d=-43).

A := Basis( ModularForms( Gamma1(43), 1), 87); A[1] + 2*A[2] + 2*A[5] + 2*A[10] + 4*A[12] + 4*A[14] + 2*A[17] + 4*A[18]; /* Michael Somos, Sep 07 2015 */

a[ n_] := If[ n < 1, Boole[n == 0], 2 DivisorSum[ n, KroneckerSymbol[ -43, #] &]]; (* Michael Somos, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^43] + EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^43], {q, 0, n}]; (* Michael Somos, Sep 07 2015 *)
Join[{1}, a[n_]:=If[n<0, 0, DivisorSum[n, KroneckerSymbol[-43, #]&]];
2 Table[a[n], {n, 1, 100}]] (* Vincenzo Librandi, Sep 07 2018 *)

{a(n) = if( n<1, n==0, sumdiv(n, d, kronecker(-43, d))*2)};

{a(n) = if( n<0, 0, polcoeff( 1 + 2 * x * Ser(qfrep([2, 1; 1, 22], n, 1)), n))};

a(n)=if(n, sumdivmult(n,d,kronecker(-43,d))*2, 0) \\ Charles R Greathouse IV, Nov 23 2021

Expansion of theta_3(q) * theta_3(q^43) + theta_2(q) * theta_2(q^43) in powers of q.
Expansion of phi(q) * phi(q^43) + 4 * q^11 * psi(q^2) * psi(q^86) in powers of q where phi(), psi() are Ramanujan theta functions.
Moebius transform is period 43 sequence [2, -2, -2, 2, -2, 2, -2, -2, 2, 2, 2, -2, 2, 2, 2, 2, 2, -2, -2, -2, 2, -2, 2, 2, 2, -2, -2, -2, -2, -2, 2, -2, -2, -2, 2, 2, -2, 2, -2, 2, 2, -2, 0, ...].
a(n) = 2*b(n) where b() is multiplicative with b(43^e) = 1, b(p^e) = e + 1 if Kronecker(-43, p) = 1, b(p^e) = (1 + (-1)^e) / 2 otherwise.
G.f. is a period 1 Fourier series which satisfies f(-1 / (43 t)) = 43^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
a(4*n + 2) = a(9*n + 3) = a(9*n + 6) = 0. a(4*n) = a(9*n) = a(n).
G.f.: Sum_{i,j in Z} x^(i*i + i*j + 11*j*j).
a(n) = 2 * A035147(n) unless n = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/sqrt(43) = 0.958176... . - Amiram Eldar, Nov 21 2023

A320234 Expansion of Product_{k=1..8} theta_3(q^k), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 2, 6, 8, 10, 22, 26, 36, 58, 72, 96, 130, 164, 200, 268, 324, 376, 486, 552, 642, 796, 876, 992, 1198, 1294, 1436, 1682, 1794, 1964, 2268, 2428, 2556, 2980, 3116, 3304, 3876, 3940, 4252, 4896, 4996, 5348, 6164, 6260, 6668, 7686, 7808, 8120, 9378, 9490, 9762

Offset: 0

Seiichi Manyama, Oct 08 2018

Also the number of integer solutions (a_1, a_2, ... , a_8) to the equation a_1^2 + 2*a_2^2 + ... + 8*a_8^2 = n.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Product_{k=1..m} theta_3(q^k): A000122 (m=1), A033715 (m=2), A029594 (m=3), A320139 (m=4), A320231 (m=5), A320232 (m=6), A320233 (m=7), this sequence (m=8), A320241 (m=9), A320242(m=10), A320246 (m=12), A320247 (m=16).

Cf. A320067, A320243.

A320231 Expansion of Product_{k=1..5} theta_3(q^k), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 2, 6, 8, 10, 20, 20, 26, 38, 40, 48, 54, 60, 56, 80, 76, 60, 106, 76, 102, 132, 100, 128, 160, 174, 136, 210, 164, 164, 280, 160, 182, 256, 216, 232, 320, 204, 244, 408, 288, 288, 368, 316, 292, 518, 276, 264, 510, 310, 454, 480, 380, 408, 616, 524, 428, 656

Offset: 0

Seiichi Manyama, Oct 08 2018

nonn

Also the number of integer solutions (a_1, a_2, ... , a_5) to the equation a_1^2 + 2*a_2^2 + ... + 5*a_5^2 = n.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Product_{k=1..m} theta_3(q^k): A000122 (m=1), A033715 (m=2), A029594 (m=3), A320139 (m=4), this sequence (m=5), A320232 (m=6), A320233 (m=7), A320234 (m=8).

Cf. A320067.

A320232 Expansion of Product_{k=1..6} theta_3(q^k), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 2, 6, 8, 10, 22, 24, 30, 50, 56, 68, 94, 100, 108, 156, 156, 156, 214, 196, 214, 292, 252, 248, 374, 330, 344, 486, 380, 440, 640, 548, 506, 752, 624, 656, 988, 644, 720, 1080, 872, 872, 1220, 876, 984, 1598, 1052, 1096, 1566, 1290, 1310, 1936, 1260, 1264, 2198

Offset: 0

Seiichi Manyama, Oct 08 2018

nonn

Also the number of integer solutions (a_1, a_2, ... , a_6) to the equation a_1^2 + 2*a_2^2 + ... + 6*a_6^2 = n.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Product_{k=1..m} theta_3(q^k): A000122 (m=1), A033715 (m=2), A029594 (m=3), A320139 (m=4), A320231 (m=5), this sequence (m=6), A320233 (m=7), A320234 (m=8).

Cf. A320067.

A320233 Expansion of Product_{k=1..7} theta_3(q^k), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 2, 6, 8, 10, 22, 26, 34, 54, 68, 84, 114, 144, 156, 216, 256, 268, 350, 384, 414, 508, 564, 560, 686, 758, 736, 914, 966, 948, 1140, 1308, 1182, 1460, 1640, 1464, 1928, 2024, 1928, 2228, 2564, 2320, 2748, 3164, 2584, 3350, 3640, 3232, 3738, 4314, 3566, 4400

Offset: 0

Seiichi Manyama, Oct 08 2018

Also the number of integer solutions (a_1, a_2, ... , a_7) to the equation a_1^2 + 2*a_2^2 + ... + 7*a_7^2 = n.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Product_{k=1..m} theta_3(q^k): A000122 (m=1), A033715 (m=2), A029594 (m=3), A320139 (m=4), A320231 (m=5), A320232 (m=6), this sequence (m=7), A320234 (m=8).

Cf. A320067.

A082564 Expansion of eta(q)^2 * eta(q^2) / eta(q^4) in powers of q.

Original entry on oeis.org

1, -2, -2, 4, 2, 0, -4, 0, 2, -6, 0, 4, 4, 0, 0, 0, 2, -4, -6, 4, 0, 0, -4, 0, 4, -2, 0, 8, 0, 0, 0, 0, 2, -8, -4, 0, 6, 0, -4, 0, 0, -4, 0, 4, 4, 0, 0, 0, 4, -2, -2, 8, 0, 0, -8, 0, 0, -8, 0, 4, 0, 0, 0, 0, 2, 0, -8, 4, 4, 0, 0, 0, 6, -4, 0, 4, 4, 0, 0, 0, 0, -10, -4, 4, 0, 0, -4, 0, 4, -4, 0, 0, 0, 0, 0, 0, 4, -4, -2, 12, 2, 0, -8, 0

Offset: 0

Benoit Cloitre, May 05 2003

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
a(n) is nonzero if and only if n is in A002479. - Michael Somos, Dec 15 2011
Absolute values appear to give A033715 = 2*A002325.
Denoted by a_4(n) in Kassel and Reutenauer 2015. - Michael Somos, Jun 04 2015

			G.f. = 1 - 2*q - 2*q^2 + 4*q^3 + 2*q^4 - 4*q^6 + 2*q^8 - 6*q^9 + 4*q^11 + 4*q^12 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015 , see page 31 7.2(c). [Note that a later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication which is next in this list.]
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016, see p. 13 paragraph 3.3.3.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002479, A033715, A033761, A125095, A129134, A133692, A258747, A258764.

A := Basis( ModularForms( Gamma1(32), 1), 105); A[1] - 2*A[2] - 2*A[3] + 4*A[4] + 2*A[5] - 4*A[7] + 2*A[9] - 6*A[10] + 4*A[12] + 4*A[13] - 4*A[16]; /* Michael Somos, Aug 29 2014 */

a[ n_] := SeriesCoefficient[ QPochhammer[ q]^2 QPochhammer[ q^2] / QPochhammer[ q^4], {q, 0, n}]; (* Michael Somos, Aug 29 2014 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^2], {q, 0, n}]; (* Michael Somos, Aug 29 2014 *)
a[ n_] := If[ n < 1, Boole[ n == 0], 2 (-1)^Quotient[ n + 1, 2] DivisorSum[ n, KroneckerSymbol[ -2, #] &]]; (* Michael Somos, Aug 29 2014 *)

{a(n) = if( n<1, n==0, 2 * (-1)^((n+1) \ 2) * sumdiv( n, d, kronecker( -2, d)))}; /* Michael Somos, Mar 30 2007 */

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

Expansion of phi(-q) * phi(-q^2) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Mar 30 2007
Euler transform of period 4 sequence [ -2, -3, -2, -2, ...]. - Michael Somos, Mar 30 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^(11/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A033761. - Michael Somos, Aug 29 2014
G.f.: Product_{k>0} (1 - x^k)^2 / (1 + x^(2*k)). - Michael Somos, Mar 30 2007
a(n) = -2 * A129134(n) unless n=0. - Michael Somos, Mar 30 2007
a(n) = (-1)^floor( (n+1)/2 ) * A033715(n). - Michael Somos, Aug 29 2014
a(2*n) = A133692(n). a(2*n + 1) = -2 * A125095(n). - Michael Somos, Aug 29 2014
a(3*n + 1) = -2 * A258747(n). a(3*n + 2) = -2 * A258764(n). - Michael Somos, Jun 09 2015

A133692 Expansion of phi(-q) * phi(q^2) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 2, -4, 2, 0, 4, 0, 2, -6, 0, -4, 4, 0, 0, 0, 2, -4, 6, -4, 0, 0, 4, 0, 4, -2, 0, -8, 0, 0, 0, 0, 2, -8, 4, 0, 6, 0, 4, 0, 0, -4, 0, -4, 4, 0, 0, 0, 4, -2, 2, -8, 0, 0, 8, 0, 0, -8, 0, -4, 0, 0, 0, 0, 2, 0, 8, -4, 4, 0, 0, 0, 6, -4, 0, -4, 4, 0, 0, 0, 0

Offset: 0

Michael Somos, Sep 20 2007

sign

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

a(n) is nonzero if and only if n is in A002479.

			G.f. = 1 - 2*q + 2*q^2 - 4*q^3 + 2*q^4 + 4*q^6 + 2*q^8 - 6*q^9 - 4*q^11 + ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002479, A033715, A113441, A133690.

A := Basis( ModularForms( Gamma1(16), 1), 80); A[1] -2*A[2] +2*A[3] - 4*A[4] + 2*A[5] + 4*A[7]; /* Michael Somos, Aug 29 2014 */

a[ n_] := If[ n < 1, Boole[n == 0], (-1)^n 2 DivisorSum[ n, KroneckerSymbol[ -2, #] &]]; (* Michael Somos, Oct 30 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^2], {q, 0, n}]; (* Michael Somos, Aug 29 2014 *)

{a(n) = if( n<1, n==0, (-1)^n * 2 * sumdiv(n, d, kronecker( -2, d)))};

{a(n) = my(A); if ( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^5 / (eta(x^2 + A)^3 * eta(x^8 + A)^2), n))};

Expansion of eta(q)^2 * eta(q^4)^5 / (eta(q^2)^3 * eta(q^8)^2) in powers of q.
Euler transform of period 8 sequence [ -2, 1, -2, -4, -2, 1, -2, -2, ...].
Moebius transform is period 16 sequence [ -2, 4, -2, 0, 2, 4, 2, 0, -2, -4, -2, 0, 2, -4, 2, 0, ...].
a(n) = -2 * b(n) where b(n) is multiplicative with b(2^e) = -1 if e>0, b(p^e) = (1 + (-1)^e) / 2 if p == 5, 7 (mod 8), b(p^e) = e + 1 if p == 1, 3 (mod 8).
G.f.: Product_{k>0} (1 - x^k)^2 * (1 + x^(2*k))^3 / (1 + x^(4*k))^2.
a(8*n + 5) = a(8*n + 7) = 0.
A133690 is the convolution square. a(n) = (-1)^n * A033715(n). a(2*n) = A033715(n). a(2*n + 1) = -2 * A113411(n).

A319822 Number of solutions to x^2 + 2y^2 + 5z^2 + 5*w^2 = n.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 12, 8, 18, 14, 4, 28, 12, 24, 32, 0, 34, 20, 14, 28, 4, 32, 44, 40, 28, 10, 40, 56, 64, 72, 8, 48, 66, 24, 68, 8, 46, 88, 60, 32, 4, 52, 64, 116, 76, 12, 64, 72, 60, 82, 26, 72, 104, 104, 88, 8, 112, 56, 136, 140, 8, 136, 96, 72, 98, 16, 72, 132

Offset: 0

Jianing Song, Sep 28 2018

nonn

Ramanujan (1917) claimed that there are exactly 55 possible choice for a <= b <= c <= d such that a*x^2 + b*y^2 + c*z^2 + d*w^2 represents all natural numbers, but L. E. Dickson (1927) has pointed out that Ramanujan has overlooked the fact that (1, 2, 5, 5) does not represent 15. Consequently, there are only 54 forms. This sequence is related to the form (1, 2, 5, 5). As is proven, a(n) = 0 iff n = 15.

There are also many (a, b, c, d) other than this such that a*x^2 + b*y^2 + c*z^2 + d*w^2 represents all but finitely many natural numbers. For example, x^2 + y^2 + 5*z^2 + 5*w^2 represents all natural numbers except for 3 (cf. A236929); x^2 + y^2 + z^2 + d*w^2 (d == 2 (mod 4) or d = 9, 17, 25, 36, 68, 100 and some others) represents all natural numbers except for those of the form 4^i*(8*j + 7) and < d; x^2 + 2*y^2 + 6*z^2 + d*w^2 (d == 2 (mod 4) or d = 11, 19 and some others) represents all natural numbers except for those of the form 4^i*(8*j + 5) and < d.

			a(5) = 4 because 0^2 + 2*0^2 + 5*0^2 + 5*1^2 = 0^2 + 2*0^2 + 5*0^2 + 5*(-1)^2 = 0^2 + 2*0^2 + 5*1^2 + 5*0^2 = 0^2 + 2*0^2 + 5*(-1)^2 + 5*0^2 = 5 and these are the only four solutions to x^2 + 2*y^2 + 5*z^2 + 5*w^2 = 5.

J. H. Conway, Universal quadratic forms and the fifteen theorem, Contemporary Mathematics 272 (1999), 23-26.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
L. E. Dickson, Integers represented by positive ternary quadratic forms, Bulletin of the American Mathematical Society, 1927, 33(1):63-70.
H. D. Kloosterman, On the representation of numbers in the form ax^2 + by^2 + cz^2 + dt^2, Acta Mathematica, 1927, 49(3-4):407-464.
S. Ramanujan, On the expression of a number in the form ax^2 + by^2 + cz^2 + du^2, Proc. Camb. Phil. Soc. 19 (1917), 11-21.

Cf. A004018, A033715, A236922-A236933.
From Seiichi Manyama, Oct 07 2018: (Start)
possible choice:
  k   |   a, b, c,  d   | Number of solutions
------+-----------------+--------------------
|   1, 1, 1,  1   | A000118
|   1, 1, 1,  2   | A236928
|   1, 1, 1,  3   | A236926
|   1, 1, 1,  4   | A236923
|   1, 1, 1,  5   | A236930
|   1, 1, 1,  6   | A236931
|   1, 1, 1,  7   | A236932
|   1, 1, 2,  2   | A097057
|   1, 1, 2,  3   | A320124
|   1, 1, 2,  4   | A320125
|   1, 1, 2,  5   | A320126
|   1, 1, 2,  6   | A320127
|   1, 1, 2,  7   | A320128
|   1, 1, 2,  8   | A320130
|   1, 1, 2,  9   | A320131
|   1, 1, 2, 10   | A320132
|   1, 1, 2, 11   | A320133
|   1, 1, 2, 12   | A320134
|   1, 1, 2, 13   | A320135
|   1, 1, 2, 14   | A320136
|   1, 1, 3,  3   | A034896
|   1, 1, 3,  4   | A272364
|   1, 1, 3,  5   | A320147
|   1, 1, 3,  6   | A320148
|   1, 2, 2,  2   | A320149
|   1, 2, 2,  3   | A320150
|   1, 2, 2,  4   | A236924
|   1, 2, 2,  5   | A320151
|   1, 2, 2,  6   | A320152
|   1, 2, 2,  7   | A320153
|   1, 2, 3,  3   | A320138
|   1, 2, 3,  4   | A320139
|   1, 2, 3,  5   | A320140
|   1, 2, 3,  6   | A033712
|   1, 2, 3,  7   | A320188
|   1, 2, 3,  8   | A320189
|   1, 2, 3,  9   | A320190
|   1, 2, 3, 10   | A320191
|   1, 2, 4,  4   | A320193
|   1, 2, 4,  5   | A320194
|   1, 2, 4,  6   | A320195
|   1, 2, 4,  7   | A320196
|   1, 2, 4,  8   | A033720
|   1, 2, 4,  9   | A320197
|   1, 2, 4, 10   | A320198
|   1, 2, 4, 11   | A320199
|   1, 2, 4, 12   | A320200
|   1, 2, 4, 13   | A320201
|   1, 2, 4, 14   | A320202
|   1, 2, 5,  6   | A320163
|   1, 2, 5,  7   | A320164
|   1, 2, 5,  8   | A320165
|   1, 2, 5,  9   | A320166
|   1, 2, 5, 10   | A033722
(End)

JT := (k, n) -> JacobiTheta3(0, x^k)^n:
A319822List := proc(len) series(JT(1,1)*JT(2,1)*JT(5,2), x, len+1);
seq(coeff(%, x, j), j=0..len) end: A319822List(67); # Peter Luschny, Oct 01 2018

CoefficientList[EllipticTheta[3, 0, q] EllipticTheta[3, 0, q^2] EllipticTheta[ 3, 0, q^5]^2 + O[q]^100, q] (* Jean-François Alcover, Jun 15 2019 *)

A004018(n) = if(n, 4*sumdiv(n,d,kronecker(-4,d)), 1);
A033715(n) = if(n, 2*sumdiv(n,d,kronecker(-2,d)), 1);
a(n) = my(i=0); for(k=0, n\5, i+=A004018(k)*A033715(n-5*k)); i

N=99; q='q+O('q^N);
gf = (eta(q^2)*eta(q^4))^3*eta(q^10)^10/(eta(q)*eta(q^5)^2*eta(q^8)*eta(q^20)^2)^2;
Vec(gf) \\ Altug Alkan, Oct 01 2018

Q = DiagonalQuadraticForm(ZZ, [1, 2, 5, 5])
Q.theta_series(68).list() # Peter Luschny, Oct 01 2018

a(n) = Sum_{k=0..floor(n/5)} A004018(k)*A033715(n-5*k).

G.f.: theta_3(q)*theta_3(q^2)*theta_3(q^5)^2, where theta_3() is the Jacobi theta function.

A320241 Expansion of Product_{k=1..9} theta_3(q^k), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 2, 6, 8, 10, 22, 26, 36, 60, 76, 100, 142, 180, 220, 312, 376, 448, 602, 696, 834, 1056, 1204, 1392, 1734, 1942, 2188, 2654, 2898, 3248, 3860, 4180, 4540, 5376, 5704, 6176, 7242, 7532, 8184, 9444, 9868, 10480, 12168, 12544, 13348, 15554, 15832, 16816, 19430

Offset: 0

Seiichi Manyama, Oct 08 2018

Also the number of integer solutions (a_1, a_2, ... , a_9) to the equation a_1^2 + 2*a_2^2 + ... + 9*a_9^2 = n.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Product_{k=1..m} theta_3(q^k): A000122 (m=1), A033715 (m=2), A029594 (m=3), A320139 (m=4), A320231 (m=5), A320232 (m=6), A320233 (m=7), A320234 (m=8), this sequence (m=9), A320242 (m=10).

Cf. A320067.

A320242 Expansion of Product_{k=1..10} theta_3(q^k), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 2, 6, 8, 10, 22, 26, 36, 60, 78, 104, 146, 192, 236, 332, 420, 500, 674, 816, 986, 1256, 1488, 1752, 2174, 2566, 2940, 3550, 4102, 4640, 5528, 6292, 6948, 8160, 9172, 10060, 11618, 12840, 13980, 15940, 17590, 18844, 21252, 23308, 24772, 27926, 30360, 31932

Offset: 0

Seiichi Manyama, Oct 08 2018

nonn

Also the number of integer solutions (a_1, a_2, ... , a_10) to the equation a_1^2 + 2*a_2^2 + ... + 10*a_10^2 = n.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Product_{k=1..m} theta_3(q^k): A000122 (m=1), A033715 (m=2), A029594 (m=3), A320139 (m=4), A320231 (m=5), A320232 (m=6), A320233 (m=7), A320234 (m=8), A320241 (m=9), this sequence (m=10).

Cf. A320067.

cp's OEIS Frontend

A138811 Theta series of quadratic form x^2 + x*y + 11*y^2.

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A320234 Expansion of Product_{k=1..8} theta_3(q^k), where theta_3() is the Jacobi theta function.

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A320231 Expansion of Product_{k=1..5} theta_3(q^k), where theta_3() is the Jacobi theta function.

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A320232 Expansion of Product_{k=1..6} theta_3(q^k), where theta_3() is the Jacobi theta function.

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A320233 Expansion of Product_{k=1..7} theta_3(q^k), where theta_3() is the Jacobi theta function.

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A082564 Expansion of eta(q)^2 * eta(q^2) / eta(q^4) in powers of q.

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A133692 Expansion of phi(-q) * phi(q^2) in powers of q where phi() is a Ramanujan theta function.

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A319822 Number of solutions to x^2 + 2*y^2 + 5*z^2 + 5*w^2 = n.

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A138811 Theta series of quadratic form x^2 + xy + 11y^2.

A319822 Number of solutions to x^2 + 2y^2 + 5z^2 + 5*w^2 = n.