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A002300
Coefficients in the expansion of B^2*C^3 in Watson's notation of page 118.
(Formerly M0093 N0029)
3
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
0,2
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).
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).
LINKS
Watson, G. N., Ramanujans Vermutung ueber Zerfaellungsanzahlen. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128. See p. 128.
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
EXAMPLE
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 + ...
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 *)
PROG
(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 */
CROSSREFS
Cf. A160553.
Sequence in context: A161283 A226516 A366844 * A350063 A049099 A181776
KEYWORD
sign,easy
EXTENSIONS
Entry revised by N. J. A. Sloane, Nov 14 2009
STATUS
approved