A292420 - OEIS

%I #22 Sep 17 2017 13:35:58

%S 1,2,2,3,4,4,6,8,8,11,14,16,20,24,28,34,42,48,57,68,78,94,110,126,148,

%T 172,198,230,266,304,351,404,460,526,602,684,780,888,1004,1140,1290,

%U 1456,1646,1856,2088,2351,2644,2964,3326,3728,4168,4664,5212,5812,6484

%N Expansion of a q-series used by Ramanujan in his Lost Notebook.

%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, page 1, 1st equation with a=-1.

%H Robert Israel, <a href="/A292420/b292420.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f. is the product of the g.f. of A000009 and A143064.

%F Given g.f. A(x), then A(x^2) = 1 / (1+x) + x / (1+x^3) + x^2 * (1+x^2) / ((1+x^3) * (1+x^5)) + x^3 * (1+x^2) / ((1+x^5) * (1+x^7)) + x^4 * (1+x^2) * (1+x^4) / ((1+x^5) * (1+x^7) * (1+x^9)) + ...

%e G.f. = 1 + 2*x + 2*x^2 + 3*x^3 + 4*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 8*x^8 + ...

%p N:= 200: # to get a(0)..a(N)

%p g143064:= add(x^k/mul(1+x^(2*j+1),j=0..k),k=0..2*N):

%p g000009:= mul(1+x^(2*k),k=1..N):

%p S:= series(g143064*g000009,x,2*N+2):

%p seq(coeff(S,x,2*j),j=0..N); # _Robert Israel_, Sep 17 2017

%t a[ n_] := If[ n < 0, 0, SeriesCoefficient[ QPochhammer[ x^2] / QPochhammer[ x] Sum[ (-1)^k x^(3 k^2 + 2 k) (1 + x^(2 k + 1)), {k, 0, Sqrt[n / 3]}], {x, 0, n}]];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) / eta(x + A) * sum(k=0, sqrtint(n \ 3), (-1)^k * x^(3*k^2 + 2*k) * (1 + x^(2*k + 1)), A), n))};

%Y Cf. A000009, A143064, A292445.

%K nonn

%O 0,2

%A _Michael Somos_, Sep 16 2017