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 A292420 Expansion of a q-series used by Ramanujan in his Lost Notebook. 2
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 REFERENCES Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, page 1, 1st equation with a=-1.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

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Last modified June 6 03:09 EDT 2023. Contains 363138 sequences. (Running on oeis4.)