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

0,2

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, page 1, 1st equation with a=-1.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

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)) + ...

Example

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 + ...

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}]];

Prog

(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))};

Crossrefs

Cf. A000009, A143064, A292445.

Sequence in context: A173673 A330715 A018125 * A161654 A325854 A301370

Adjacent sequences: A292417 A292418 A292419 * A292421 A292422 A292423

Keyword

nonn

Author

Michael Somos, Sep 16 2017

Status

approved