A292420 Expansion of a q-series used by Ramanujan in his Lost Notebook.

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, 172, 198, 230, 266, 304, 351, 404, 460, 526, 602, 684, 780, 888, 1004, 1140, 1290, 1456, 1646, 1856, 2088, 2351, 2644, 2964, 3326, 3728, 4168, 4664, 5212, 5812, 6484

Offset: 0

Author

Michael Somos, Sep 16 2017

Keywords

• nonn

Examples

			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 + ...
		

References

• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, page 1, 1st equation with a=-1.

Links

• Robert Israel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000009, A143064, A292445.

Programs

Maple

N:= 200: # to get a(0)..a(N)
g143064:= add(x^k/mul(1+x^(2*j+1),j=0..k),k=0..2*N):
g000009:= mul(1+x^(2*k),k=1..N):
S:= series(g143064*g000009,x,2*N+2):
seq(coeff(S,x,2*j),j=0..N); # Robert Israel, Sep 17 2017

Mathematica

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}]];

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))};

Formula

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

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)) + ...