cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-18 of 18 results.

A120030 Expansion of theta_4(q)^2*theta_4(q^2)^4 in powers of q.

Original entry on oeis.org

1, -4, -4, 32, -4, -104, 32, 192, -4, -292, -104, 480, 32, -680, 192, 832, -4, -1160, -292, 1440, -104, -1536, 480, 2112, 32, -2604, -680, 2624, 192, -3368, 832, 3840, -4, -3840, -1160, 4992, -292, -5480, 1440, 5440, -104, -6728, -1536, 7392, 480, -7592, 2112, 8832, 32, -9412, -2604
Offset: 0

Views

Author

Michael Somos, Jun 05 2006

Keywords

Comments

Number 8 of the 74 eta-quotients listed in Table I of Martin (1996).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 - 4*q - 4*q^2 + 32*q^3 - 4*q^4 - 104*q^5 + 32*q^6 + 192*q^7 - 4*q^8 + ...

References

  • N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.7).

Links

Crossrefs

Cf. A000141, A002173, A050470, A128692.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(4), 3), 51); A[1] - 4*A[2]; /* Michael Somos, May 24 2015 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2]^3 (QPochhammer[ q] / QPochhammer[ q^4])^2)^2, {q, 0, n}]; (* Michael Somos, May 24 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], -4 DivisorSum[ n, #^2 KroneckerSymbol[ -4, #] &]]; (* Michael Somos, May 24 2015 *)
  • PARI
    {a(n) = if( n<1, n==0, -4 * sumdiv( n, d, d^2 * kronecker( -4, d)))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^2 + A)^6 / eta(x^4 + A)^4, n))};

Formula

Expansion of eta(q)^4 * eta(q^2)^6 / eta(q^4)^4 in powers of q.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^2 * (u - v)^2 - 4 * u*w * (v - w) * (u - 2*v).
Euler transform of period 4 sequence [ -4, -10, -4, -6, ...].
G.f.: 1 - 4 * Sum_{k>0} A056594(k-1) * k^2 * x^k / (1 - x^k).
Expansion of phi(-q)^2 * phi(-q^2)^4 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Aug 15 2007
G.f.: (Sum_{k in Z} (-1)^k * x^k^2)^2 * (Sum_{k in Z} (-1)^k * x^(2*k^2))^4.
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 128 (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A050470.
a(n) = -4 * A002173(n) unless n=0.
Convolution of A000141 and A128692.

A008425 Theta series of {D_6}* lattice.

Original entry on oeis.org

1, 0, 12, 64, 60, 0, 160, 384, 252, 0, 312, 960, 544, 0, 960, 1664, 1020, 0, 876, 2880, 1560, 0, 2400, 4224, 2080, 0, 2040, 5248, 3264, 0, 4160, 7680, 4092, 0, 3480, 9984, 4380, 0, 7200, 10880, 6552, 0, 4608, 14784, 8160, 0, 10560, 17664, 8224, 0, 7812, 18560
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Examples

			G.f. = 1 + 12*x^2 + 64*x^3 + 60*x^4 + 160*x^6 + 384*x^7 + 252*x^8 + 312*x^10 + ...

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag New York, 1999, ISBN 978-1-4757-6568-7, p. 120.

Links

Crossrefs

Cf. A000141, A008428.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(8), 3), 52); A[1] + 12*A[3] + 64*A[4] + 60*A[5] + 160*A[7]; /* Michael Somos, Dec 14 2016 */
  • Mathematica
    a[n_] := DivisorSum[n, #^2*(4*(KroneckerSymbol[-4, n/#]-KroneckerSymbol[-4, #]))&]; a[0]=1; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 06 2016, after Ralf Stephan *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^2]^6 + EllipticTheta[ 2, 0, x^2]^6, {x, 0, n}]; (* Michael Somos, Dec 14 2016 *)
  • PARI
    {a(n) = if( n<1, n==0, 4 * sumdiv(n, d, d^2 * (kronecker(-4, n/d) - kronecker(-4, d))))}; /* Michael Somos, Dec 14 2016 */
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A)^5 / (eta(x^2 + A) * eta(x^8 + A))^2)^6 + 64 * x^3 * (eta(x^8 + A)^2/ eta(x^4 + A))^6, n))}; /* Michael Somos, Dec 14 2016 */
  • PARI
    {a(n) = my(G); if( n<0, 0, G = [2, 0, 0, 0, 0, 1; 0, 2, 0, 0, 0, 1; 0, 0, 2, 0, 0, 1; 0, 0, 0, 2, 0, 1; 0, 0, 0, 0, 2, 1; 1, 1, 1, 1, 1, 3]; polcoeff( 1 + 2 * x * Ser( qfrep( G, n)), n))}; /* Michael Somos, Dec 14 2016 */

Formula

Apparently, a(n) = Sum_{d|n} d^2*(4*(Kronecker(-4,n/d) - Kronecker(-4,d))), n > 0. - Ralf Stephan, Dec 31 2014
Expansion of phi(x^2)^6 + 64 * x^3 * psi(x^4)^6 in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Dec 14 2016
G.f. is a period 1 Fourier series that satisfies f(-1 / (8 t)) = 16 (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A008428. - Michael Somos, Dec 14 2016
G.f.: theta_3(0, x^2)^6 + theta_2(0, x^2)^6.

A008428 Theta series of D_6 lattice.

Original entry on oeis.org

1, 60, 252, 544, 1020, 1560, 2080, 3264, 4092, 4380, 6552, 8160, 8224, 10200, 12480, 14144, 16380, 17400, 18396, 24480, 26520, 23040, 31200, 35904, 32800, 39060, 42840, 44608, 49344, 50520, 54080, 65280, 65532, 57600, 73080, 84864, 74460, 82200, 93600, 92480
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Examples

			G.f. = 1 + 60*x + 252*x^2 + 544*x^3 + 1020*x^4 + 1560*x^5 + 2080*x^6 + ...
G.f. = 1 + 60*q^2 + 252*q^4 + 544*q^6 + 1020*q^8 + 1560*q^10 + 2080*q^12 + ...

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 118.

Links

Crossrefs

Cf. A000141, A008425.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(8), 3), 80); A[1] + 60*A[3] + 252*A[5] + 544*A[7]; /* Michael Somos, Aug 26 2015 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x]^6, {x, 0, 2 n}]; (* Michael Somos, Aug 26 2015 *)
  • PARI
    {a(n) = if( n<1, n==0, 4 * sumdiv(n, d, d^2 * (16 * kronecker(-4, n/d) - kronecker(-4, d))))}; /* Michael Somos, Nov 03 2006 */
  • PARI
    {a(n) = if( n<0, 0, n*=2; polcoeff( sum(k=1, sqrtint(n), 2*x^k^2, 1 + x * O(x^n))^6, n))}; /* Michael Somos, Nov 03 2006 */

Formula

G.f.: (theta_3(q^(1/2))^6 + theta_4(q^(1/2))^6)/2
Expansion of ( phi(q)^6 + phi(-q)^6 ) / 2 in powers of q^2 where phi() is a Ramanujan theta function. - Michael Somos, Sep 14 2007
a(n) = A000141(2*n).
G.f. is a period 1 Fourier series that satisfies f(-1 / (8 t)) = 12 (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A008425. - Michael Somos, Aug 26 2015

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

Views

Author

Hugo Pfoertner, Sep 12 2007

Keywords

Examples

			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.

Links

Crossrefs

Cf. A000161, A000378, A000141, A005875, A000118, A000132, A008451.
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).

Programs

  • PARI
    a(i, n, k)=local(s, j); if(k==1, if(issquare(n) && n

Extensions

2 more terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 16 2007
More terms from Robert Gerbicz, May 09 2008

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

Views

Author

Hugo Pfoertner, Sep 11 2007

Keywords

Examples

			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.

Links

Crossrefs

Cf. A000161, A000378, A000141, A005875, A000118, A000132, A008451.
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).

Programs

  • PARI
    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

Extensions

2 more terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 16 2007
More terms from Robert Gerbicz, May 09 2008

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

Views

Author

Hugo Pfoertner, Sep 12 2007

Keywords

Examples

			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.

Links

Crossrefs

Cf. A000161, A000378, A000141, A005875, A000118, A000132, A008451.
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).

Programs

  • PARI
    a(i, n, k)=local(s, j); if(k==1, if(issquare(n) && n

Extensions

a(10) from Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 16 2007
a(11) onwards from Robert Gerbicz, May 09 2008

A004407 Expansion of ( Sum_{n = -infinity..infinity} x^(n^2) )^(-6).

Original entry on oeis.org

1, -12, 84, -448, 2004, -7896, 28224, -93312, 289236, -848972, 2377704, -6391872, 16571968, -41599320, 101430144, -240877440, 558440916, -1266406680, 2814053908, -6136337088, 13148606184, -27717527552
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000122, A000141.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Product[((1 + (-x)^k)/(1 - (-x)^k))^6, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 18 2015 *)
  • PARI
    q='q+O('q^99); Vec(((eta(q)*eta(q^4))^2/eta(q^2)^5)^6) \\ Altug Alkan, Sep 20 2018

Formula

a(n) ~ (-1)^n * 3^(7/4)*exp(Pi*sqrt(6*n)) / (256*2^(3/4)*n^(9/4)). - Vaclav Kotesovec, Aug 18 2015
From Ilya Gutkovskiy, Sep 20 2018: (Start)
G.f.: 1/theta_3(x)^6, where theta_3() is the Jacobi theta function.
G.f.: Product_{k>=1} 1/((1 - x^(2*k))*(1 + x^(2*k-1))^2)^6. (End)

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

Views

Author

Hugo Pfoertner, Sep 11 2007

Keywords

Examples

			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.

Crossrefs

Cf. A000161, A000378, A000141, A005875, A000118, A000132, A008451.
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).

Programs

  • PARI
    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

Extensions

a(9) from Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 16 2007
a(10) onwards from Robert Gerbicz, May 09 2008
Previous Showing 11-18 of 18 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.