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Coefficients in the expansion of B^2*C^3 in Watson's notation of page 118.
(Formerly M0093 N0029)
3

%I M0093 N0029 #26 Jan 04 2019 04:04:45

%S 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,

%T -2,-2,-8,-3,10,0,-4,2,19,-14,7,-8,0,-20,-4,-1,8,-2,-15,-7,8,26,-10,

%U 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

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

%C 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.

%C Watson's main reason for computing this sequence was to study values of n such that partition(49n+47) == 0 mod 343 (cf. A160553).

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

%H N. J. A. Sloane, <a href="/A002300/b002300.txt">Table of n, a(n) for n = 0..199</a>

%H Watson, G. N., <a href="http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002174499">Ramanujans Vermutung ueber Zerfaellungsanzahlen.</a> J. Reine Angew. Math. (Crelle), 179 (1938), 97-128. See p. 128.

%F Expansion of q^(-23/24) * eta(q)^2 * eta(q^7)^3 in powers of q. - _Michael Somos_, May 31 2012

%F Euler transform of period 7 sequence [ -2, -2, -2, -2, -2, -2, -5, ...]. - _Michael Somos_, May 31 2012

%F G.f.: Product_{k>0} (1 - x^k)^2 * (1 - x^(7*k))^3. - _Michael Somos_, May 31 2012

%e 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 + ...

%e 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 + ...

%p M1:=2400:

%p fm:=mul(1-x^n,n=1..M1):

%p B:=x*subs(x=x^24,fm):

%p C:=x^7*subs(x=x^168,fm):

%p t1:=B^2*C^3;

%p t2:=series(t1,x,M1);

%p t3:=subs(x=y^(1/24),t2/x^23);

%p t4:=series(t3,y,M1/24);

%p t5:=seriestolist(t4); # A002300

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 QPochhammer[ x^7]^3, {x, 0, n}]; (* _Michael Somos_, Apr 26 2015 *)

%o (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 */

%Y Cf. A160553.

%K sign,easy

%O 0,2

%A _N. J. A. Sloane_

%E Entry revised by _N. J. A. Sloane_, Nov 14 2009