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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 (list; graph; refs; listen; history; text; internal format)
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

N. J. A. Sloane, Table of n, a(n) for n = 0..199

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: A161258 A161283 A226516 * A049099 A181776 A243036

Adjacent sequences:  A002297 A002298 A002299 * A002301 A002302 A002303

KEYWORD

sign,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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Last modified October 17 15:01 EDT 2019. Contains 328116 sequences. (Running on oeis4.)