cp's OEIS Frontend

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

%I A002300 M0093 N0029 #28 Aug 11 2025 07:16:14
%S A002300 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 A002300 -2,-2,-8,-3,10,0,-4,2,19,-14,7,-8,0,-20,-4,-1,8,-2,-15,-7,8,26,-10,
%U A002300 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 A002300 Coefficients in the expansion of B^2*C^3 in Watson's notation of page 118.
%C A002300 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 A002300 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 A002300 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002300 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002300 N. J. A. Sloane, <a href="/A002300/b002300.txt">Table of n, a(n) for n = 0..199</a>
%H A002300 Watson, G. N., <a href="https://gdz.sub.uni-goettingen.de/id/PPN243919689_0179">Ramanujans Vermutung ueber Zerfaellungsanzahlen.</a> J. Reine Angew. Math. (Crelle), 179 (1938), 97-128. See p. 128.
%F A002300 Expansion of q^(-23/24) * eta(q)^2 * eta(q^7)^3 in powers of q. - _Michael Somos_, May 31 2012
%F A002300 Euler transform of period 7 sequence [ -2, -2, -2, -2, -2, -2, -5, ...]. - _Michael Somos_, May 31 2012
%F A002300 G.f.: Product_{k>0} (1 - x^k)^2 * (1 - x^(7*k))^3. - _Michael Somos_, May 31 2012
%e A002300 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 A002300 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 A002300 M1:=2400:
%p A002300 fm:=mul(1-x^n,n=1..M1):
%p A002300 B:=x*subs(x=x^24,fm):
%p A002300 C:=x^7*subs(x=x^168,fm):
%p A002300 t1:=B^2*C^3;
%p A002300 t2:=series(t1,x,M1);
%p A002300 t3:=subs(x=y^(1/24),t2/x^23);
%p A002300 t4:=series(t3,y,M1/24);
%p A002300 t5:=seriestolist(t4); # A002300
%t A002300 a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 QPochhammer[ x^7]^3, {x, 0, n}]; (* _Michael Somos_, Apr 26 2015 *)
%o A002300 (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 A002300 Cf. A160553.
%K A002300 sign,easy
%O A002300 0,2
%A A002300 _N. J. A. Sloane_
%E A002300 Entry revised by _N. J. A. Sloane_, Nov 14 2009