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.

A002300 Coefficients in the expansion of B^2*C^3 in Watson's notation of page 118.

Original entry on oeis.org

1, -2, -1, 2, 1, 2, -2, -3, 4, 1, -5, -3, -6, 8, 3, 4, 8, -3, 0, -2, -8, -4, -4, -13, 9, 5, 18, -2, -2, -8, -3, 10, 0, -4, 2, 19, -14, 7, -8, 0, -20, -4, -1, 8, -2, -15, -7, 8, 26, -10, 26, 18, 10, -2, 10, -28, -29, 18, -20, -15, 6, -8, 8, -8, 2, 19, -1, 0, -8, -6, 28, -26, -6, 23, -1, 4, 12, 25, -36, -14, 8, 0, 18, 20, 21, -12, -3, -9, 0, -16, -48
Offset: 0

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Comments

Although Watson says these are the coefficients theta_n defined on page 128, it appears that this is a mistake, and they are really the coefficients theta'_n. The true theta_n are given in A160528.
Watson's main reason for computing this sequence was to study values of n such that partition(49n+47) == 0 mod 343 (cf. A160553).

Examples

			G.f. = 1 - 2*x - x^2 + 2*x^3 + x^4 + 2*x^5 - 2*x^6 - 3*x^7 + 4*x^8 + x^9 - 5*x^10 + ...
G.f. = q^23 - 2*q^47 - q^71 + 2*q^95 + q^119 + 2*q^143 - 2*q^167 - 3*q^191 + 4*q^215 + ...
		

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Crossrefs

Cf. A160553.

Programs

  • Maple
    M1:=2400:
    fm:=mul(1-x^n,n=1..M1):
    B:=x*subs(x=x^24,fm):
    C:=x^7*subs(x=x^168,fm):
    t1:=B^2*C^3;
    t2:=series(t1,x,M1);
    t3:=subs(x=y^(1/24),t2/x^23);
    t4:=series(t3,y,M1/24);
    t5:=seriestolist(t4); # A002300
  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 QPochhammer[ x^7]^3, {x, 0, n}]; (* Michael Somos, Apr 26 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^7 + A)^3, n))}; /* Michael Somos, May 31 2012 */

Formula

Expansion of q^(-23/24) * eta(q)^2 * eta(q^7)^3 in powers of q. - Michael Somos, May 31 2012
Euler transform of period 7 sequence [ -2, -2, -2, -2, -2, -2, -5, ...]. - Michael Somos, May 31 2012
G.f.: Product_{k>0} (1 - x^k)^2 * (1 - x^(7*k))^3. - Michael Somos, May 31 2012

Extensions

Entry revised by N. J. A. Sloane, Nov 14 2009