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.

Showing 1-4 of 4 results.

A128617 Expansion of q^2 * psi(q) * psi(q^15) in powers of q where psi() is a Ramanujan theta function.

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 2, 1, 0, 1, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0
Offset: 1

Views

Author

Michael Somos, Mar 13 2007

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Also the number of positive odd solutions to equation x^2 + 15y^2 = 8n. - Seiichi Manyama, May 21 2017

Examples

			G.f. = x^2 + x^3 + x^5 + x^8 + x^12 + 2*x^17 + x^18 + x^20 + 2*x^23 + x^27 + x^30 + ...

References

  • B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 377, Entry 9(i).

Links

Crossrefs

Cf. A035162.

Programs

  • Mathematica
    a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -60, #] - KroneckerSymbol[ 20, #] KroneckerSymbol[ -3, n/#] &] / 2]; (* Michael Somos, Nov 12 2015 *)
a[ n_] := SeriesCoefficient[ q^2 (QPochhammer[ q^2] QPochhammer[ q^30])^2 / (QPochhammer[ q] QPochhammer[ q^15]), {q, 0, n}]; (* Michael Somos, Nov 12 2015 *)
  • PARI
    {a(n) = if( n<1, 0, sumdiv(n, d, kronecker(-60, d) - kronecker(20, d) * kronecker(-3, n/d) )/2)};
  • PARI
    {a(n) = my(A); if( n<2, 0, n-=2; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^30 + A))^2 / (eta(x + A) * eta(x^15 + A)), n))};

Formula

Expansion of (eta(q^2) * eta(q^30))^2 / (eta(q) * eta(q^15)) in powers of q.
Euler transform of period 30 sequence [ 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 2, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -2, ...].
For n>0, n in A028955 equivalent to a(n) nonzero. If a(n) nonzero, a(n) = A082451(n) and a(n) = -A121362(n).
a(n)= (A082451(n) - A121362(n) )/2.
G.f.: x^2 * Product_{k>0} (1 - x^k) * (1 - x^(15*k)) * (1 + x^(2*k))^2 * (1 + x^(30*k))^2.

A106406 Expansion of (eta(q) * eta(q^15))^2 / (eta(q^3) * eta(q^5)) in powers of q.

Original entry on oeis.org

1, -2, -1, 3, -1, 2, 0, -4, 1, 2, 0, -3, 0, 0, 1, 5, -2, -2, 2, -3, 0, 0, -2, 4, 1, 0, -1, 0, 0, -2, 2, -6, 0, 4, 0, 3, 0, -4, 0, 4, 0, 0, 0, 0, -1, 4, -2, -5, 1, -2, 2, 0, -2, 2, 0, 0, -2, 0, 0, 3, 2, -4, 0, 7, 0, 0, 0, -6, 2, 0, 0, -4, 0, 0, -1, 6, 0, 0, 2
Offset: 1

Views

Author

Michael Somos, May 02 2005

Keywords

Comments

Number 30 of the 74 eta-quotients listed in Table I of Martin (1996).

Examples

			G.f. = q - 2*q^2 - q^3 + 3*q^4 - q^5 + 2*q^6 - 4*q^8 + q^9 + 2*q^10 + ...

Links

Crossrefs

Cf. A028955, A028957, A035175, A123864.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(15), 1), 80); A[2] - 2*A[3] - A[4] + 3*A[5] - A[6] + 2*A[7]; /* Michael Somos, May 18 2015 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^15])^2 / (QPochhammer[ q^3] QPochhammer[ q^5]), {q, 0, n}]; (* Michael Somos, May 18 2015 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ #, 3] KroneckerSymbol[ n/#, 5] &]]; (* Michael Somos, May 18 2015 *)
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^15 + A)^2 / (eta(x^3 + A) * eta(x^5 + A)), n))};
  • PARI
    {a(n) = if( n<1, 0, sumdiv( n, d, kronecker( d, 3) * kronecker( n/d, 5)))};
  • PARI
    {a(n) = my(A, p, e, x); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==3 || p==5, (-1)^e, (p%15) != 2^(x = valuation( p%15, 2)), (e+1)%2, (e+1) * (-1)^(x*e))))};
  • PARI
    {a(n) = if( n<1, 0, (qfrep([2, 1;1, 8],n, 1) - qfrep([4, 1;1, 4], n, 1))[n])}; /* Michael Somos, Aug 25 2006 */

Formula

Euler transform of period 15 sequence [-2, -2, -1, -2, -1, -1, -2, -2, -1, -1, -2, -1, -2, -2, -2, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = -v^3 + 4 * u*v*w + 2 * u*w^2 + u^2*w.
a(n) is multiplicative with a(3^e) = a(5^e) = (-1)^e, a(p^e) = (1 + (-1)^e) / 2 if p == 7, 11, 13, 14 (mod 15), a(p^e) = e+1 if p == 1, 4 (mod 15), a(p^e) = (e+1) * (-1)^e if p == 2, 8 (mod 15). - Michael Somos, Oct 19 2005
G.f.: (1/2) * (Sum_{n,m in Z} x^(n^2 + n*m + 4*m^2) - x^(2*n^2 + n*m + 2 *m^2)). - Michael Somos, Aug 25 2006
G.f.: Sum_{k>0} Kronecker(k, 3) * x^k * (1 - x^k) * (1 - x^(2*k)) / (1 - x^(5*k)) = Sum_{k>0} Kronecker(k, 5) * x^k * (1 - x^k) / (1 - x^(3*k)).
G.f.: x * Product_{k>0} ((1 - x^k) * (1 - x^(15*k)))^2 / ((1 - x^(3*k)) * (1 - x^(5*k))).
a(15*n + 7) = a(15*n + 11) = a(15*n + 13) = a(15*n + 14) = 0. a(3*n) = a(5*n) = -a(n).
A035175(n) = |a(n)|. a(n)>0 iff n in A028957. a(n)<0 iff n in A028955.
G.f. is a period 1 Fourier series which satisfies f(-1 / (15 t)) = 15^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, May 18 2015

A106858 Primes of the form 2x^2+xy+2y^2 with x and y nonnegative.

Original entry on oeis.org

2, 5, 23, 83, 107, 137, 173, 257, 293, 347, 353, 467, 503, 617, 647, 653, 743, 797, 857, 953, 983, 1223, 1277, 1283, 1307, 1427, 1487, 1493, 1523, 1553, 1637, 1787, 1877, 1913, 1997, 2003, 2027, 2213, 2237, 2243, 2393, 2423, 2447, 2657, 2663
Offset: 1

Views

Author

T. D. Noe, May 09 2005

Keywords

Comments

Discriminant=-15.

Links

Crossrefs

Cf. A028955, A106859, A243173, A033212.

Programs

  • Mathematica
    QuadPrimes2[a_, b_, c_, lmt_] := Module[{p, d, lst = {}, xMax, yMax}, d = b^2 - 4a*c; If[a > 0 && c > 0 && d < 0, xMax = Sqrt[lmt/a]*(1+Abs[b]/Floor[Sqrt[-d]])]; Do[ If[ 4c*lmt + d*x^2 >= 0, yMax = ((-b)*x + Sqrt[4c*lmt + d*x^2])/(2c), yMax = 0 ]; Do[p = a*x^2 + b*x*y + c*y^2; If[ PrimeQ[ p]  && !MemberQ[ lst, p], AppendTo[ lst, p]], {y, 0, yMax}], {x, 0, xMax}]; Sort[ lst]];
t2 = QuadPrimes2[2, 1, 2, 350000];
Length[t2]
t2[[Length[t2]]]
For[n=1, n <= 2000, n++, Print[n, " ", t2[[n]]]] (* From N. J. A. Sloane, Jun 17 2014 *)

Extensions

Replace Mma program by a correct program, recomputed and extended b-file. - N. J. A. Sloane, Jun 17 2014

A136599 Expansion of (eta(q) * eta(q^15))^3 in powers of q.

Original entry on oeis.org

1, -3, 0, 5, 0, 0, -7, 0, 0, 0, 9, 0, 0, 0, 0, -14, 9, 0, -15, 0, 0, 34, 0, 0, 0, -27, 0, 0, -15, 0, 33, 0, 0, 0, 0, 0, -22, 0, 0, 0, 0, 0, 0, 45, 0, -14, -15, 0, 25, 0, 0, -86, 0, 0, 0, 66, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 42, 0, 0, 0, -63, 0, 0, -75, 0, 0, 0, 0
Offset: 2

Views

Author

Michael Somos, Jan 11 2008

Keywords

Examples

			G.f. = q^2 - 3*q^3 + 5*q^5 - 7*q^8 + 9*q^12 - 14*q^17 + 9*q^18 - 15*q^20 + ...

Crossrefs

Cf. A028955, A030220, A115155.

Programs

  • Magma
    A := Basis( CuspForms( Gamma1(15), 3), 80); A[2] - 3*A[3] + 5*A[5] - 7*A[8]; /* Michael Somos, Oct 13 2015 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ (QPochhammer[ q] QPochhammer[ q^15])^3, {q, 0, n}]; (* Michael Somos, Oct 13 2015 *)
  • PARI
    {a(n) = my(A); if( n<2, 0, n -= 2; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^15 + A))^3, n))};

Formula

Euler transform of period 15 sequence [ -3, -3, -3, -3, -3, -3, -3, -3, -3, -3, -3, -3, -3, -3, -6, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (15 t)) = 15^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t).
a(n) nonzero or n=0 if and only if n is in A028955.
G.f.: x^2 * (Product_{k>0} (1 - x^k) * (1 - x^(15*k)))^3.
a(3*n) = -3 * A030220(n). a(3*n + 1) = 0. - Michael Somos, Oct 13 2015
A115155(n) = a(n) + A030220(n). - Michael Somos, Oct 13 2015
Showing 1-4 of 4 results.

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

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