A160525 - OEIS

%I #23 Sep 24 2019 02:22:01

%S 1,2,5,10,20,36,65,109,183,295,471,732,1129,1705,2554,3769,5517,7979,

%T 11458,16289,23007,32227,44869,62028,85284,116530,158432,214228,

%U 288348,386224,515156,684109,904963,1192353,1565383,2047642,2669591,3468797,4493351,5802533

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

%H Seiichi Manyama, <a href="/A160525/b160525.txt">Table of n, a(n) for n = 0..1000</a>

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

%F See Maple code for formula.

%F G.f.: Product_{n>=1} (1 - x^(7*n))/(1 - x^n)^2. - _Seiichi Manyama_, Nov 06 2016

%F a(n) ~ sqrt(13/3) * exp(sqrt(26*n/21)*Pi) / (28*n). - _Vaclav Kotesovec_, Apr 13 2017

%e G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 20*x^4 + 36*x^5 + 65*x^6 + 109*x^7 + ...

%e G.f. = q^5 + 2*q^29 + 5*q^53 + 10*q^77 + 20*q^101 + 36*q^125 + 65*q^149 + 109*q^173 + ...

%p M1:=1200:

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

%p A:=x^(1/7)*subs(x=x^(24/7),fm):

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

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

%p t1:=C/B^2;

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

%p t3:=subs(x=y^(1/24),t2/x^5);

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

%p t5:=seriestolist(t4); # A160525

%t nmax = 50; CoefficientList[Series[Product[(1 - x^(7*k))/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Apr 13 2017 *)

%Y Cf. A160526, A160527, A160528.

%Y Cf. Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^2: A000041 (k=1), A015128 (k=2), A278690 (k=3), A160461 (k=5), this sequence (k=7).

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Nov 13 2009