A008429
Theta series of D_7 lattice.
Original entry on oeis.org
1, 84, 574, 1288, 3444, 4424, 9240, 11088, 18494, 19740, 34440, 31304, 52808, 52248, 74048, 71120, 110964, 94864, 145222, 132888, 181384, 163856, 249480, 201040, 295960, 264684, 346696, 314272, 454608, 352520, 518336, 452256, 591934, 517216
Offset: 0
1 + 84*q^2 + 574*q^4 + 1288*q^6 + 3444*q^8 + ...
- J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 118.
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terms = 34; s = 1/2 (EllipticTheta[3, 0, q]^7 + EllipticTheta[4, 0, q]^7) + O[q]^(2 terms); CoefficientList[s, q^2] (* Jean-François Alcover, Jul 07 2017 *)
-
{a(n)=if(n<0, 0, n*=2; polcoeff( sum(k=1, sqrtint(n), 2*x^k^2, 1+x*O(x^n))^7, n))} /* Michael Somos, Nov 03 2006 */
A133102
Number of partitions of n^3 into n distinct nonzero squares.
Original entry on oeis.org
1, 0, 0, 0, 0, 3, 5, 20, 56, 112, 268, 618, 1922, 8531, 29021, 100407, 321531, 899618, 2937312, 9295401, 31615059, 117365818, 403433963, 1417579281, 4848439367, 15960316056, 55180971700, 190251417034, 670818005444, 2429973932322
Offset: 1
a(6) = 3 because there are 3 ways to express 6^3 = 216 as a sum of 6 distinct nonzero squares: 216 = 1^2 + 2^2 + 4^2 + 5^2 + 7^2 + 11^2 = 1^2 + 3^2 + 5^2 + 6^2 + 8^2 + 9^2 = 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 9^2.
Cf.
A133103 (number of ways to express n^3 as a sum of n nonzero squares),
A133105 (number of ways to express n^4 as a sum of n distinct nonzero squares).
-
a(i, n, k)=local(s, j); if(k==1, if(issquare(n) && n
2 more terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 16 2007
A133103
Number of partitions of n^3 into n nonzero squares.
Original entry on oeis.org
1, 1, 2, 1, 10, 34, 156, 734, 3599, 18956, 99893, 548373, 3078558, 17510598, 101960454, 599522778, 3565904170, 21438347021, 129905092421, 794292345434, 4890875249113, 30326545789640, 189195772457341, 1187032920371427
Offset: 1
a(2)=1 because the only way to express 2^3 = 8 as a sum of two squares is 8 = 2^2 + 2^2.
a(3)=2 because 3^3 = 27 = 1^2 + 1^2 + 5^2 = 3^2 + 3^2 + 3^2.
Cf.
A133102 (number of ways to express n^3 as a sum of n distinct nonzero squares),
A133104 (number of ways to express n^4 as a sum of n nonzero squares).
-
a(i, n, k)=local(s, j); if(k==1, if(issquare(n), return(1), return(0)), s=0; for(j=ceil(sqrt(n/k)), min(i, floor(sqrt(n-k+1))), s+=a(j, n-j^2, k-1)); return(s)) for(n=1,50, m=n^3; k=n; print1(a(m, m, k)", ") ) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 16 2007
2 more terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 16 2007
A133105
Number of partitions of n^4 into n distinct nonzero squares.
Original entry on oeis.org
1, 0, 1, 0, 21, 266, 2843, 55932, 884756, 13816633, 283194588, 5375499165, 125889124371, 3202887665805, 80542392920980, 2270543992935431, 64253268814048352, 1892633465941308859, 59116753827795287519, 1886846993941912938452
Offset: 1
a(3)=1 because there is exactly one way to express 3^4 as the sum of 3 distinct nonzero squares: 81 = 1^2 + 4^2 + 8^2.
Cf.
A133104 (number of ways to express n^4 as a sum of n nonzero squares),
A133102 (number of ways to express n^3 as a sum of n distinct nonzero squares).
-
a(i, n, k)=local(s, j); if(k==1, if(issquare(n) && n
a(10) from Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 16 2007
A004408
Expansion of ( Sum_{n = -infinity..infinity} x^(n^2) )^(-7).
Original entry on oeis.org
1, -14, 112, -672, 3346, -14560, 57120, -206208, 694960, -2209774, 6683040, -19345760, 53874912, -144936288, 377965760, -958231680, 2367566866, -5713057728, 13488657168, -31210552800, 70873262880, -158145658560
Offset: 0
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nmax = 30; CoefficientList[Series[Product[((1 + (-x)^k)/(1 - (-x)^k))^7, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 18 2015 *)
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q='q+O('q^99); Vec(((eta(q)*eta(q^4))^2/eta(q^2)^5)^7) \\ Altug Alkan, Sep 20 2018
A133104
Number of partitions of n^4 into n nonzero squares.
Original entry on oeis.org
1, 0, 3, 1, 49, 732, 9659, 190169, 3225654, 61896383, 1360483727, 30969769918, 778612992660, 20749789703573, 579672756740101, 17115189938667708, 525530773660159970, 16825686497823918869, 561044904645283065043, 19368002907483932784642
Offset: 1
a(3)=3 because there are 3 ways to express 3^4 = 81 as a sum of 3 nonzero squares: 81 = 1^2 + 4^2 + 8^2 = 3^2 + 6^2 + 6^2 = 4^2 + 4^2 + 7^2.
a(4)=1 because the only way to express 4^4 = 256 as a sum of 4 nonzero squares is 256 = 8^2 + 8^2 + 8^2 + 8^2.
Cf.
A133105 (number of ways to express n^4 as a sum of n distinct nonzero squares),
A133103 (number of ways to express n^3 as a sum of n nonzero squares).
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a(i, n, k)=local(s, j); if(k==1, if(issquare(n), return(1), return(0)), s=0; for(j=ceil(sqrt(n/k)), min(i, floor(sqrt(n-k+1))), s+=a(j, n-j^2, k-1)); return(s)) for(n=1,50, m=n^4; k=n; print1(a(m, m, k)", ") ) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 16 2007
a(9) from Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 16 2007
A361695
Number of ways of writing n^2 as a sum of seven squares.
Original entry on oeis.org
1, 14, 574, 3542, 18494, 43414, 145222, 235998, 591934, 860846, 1779974, 2256422, 4678982, 5195750, 9675918, 10983742, 18942014, 19873966, 35294686, 34670454, 57349894, 59707494, 92513302, 90116222, 149759302, 135668414, 213025750, 209185718, 311753358, 287144326, 450333422
Offset: 0
-
a:= n-> coeff((sum(x^(j^2), j=-n..n))^7, x, n^2):
seq(a(n), n=0..30);
# second Maple program:
b:= proc(n, t) option remember; `if`(n=0, 1, `if`(n<0 or t<1, 0,
b(n, t-1) +2*add(b(n-j^2, t-1), j=1..isqrt(n))))
end:
a:= n-> b(n^2, 7):
seq(a(n), n=0..30);
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SquaresR[7, Range[0, 30]^2] (* Paolo Xausa, Aug 21 2025 *)