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A027348
Number of partitions of n into distinct odd parts, the least being congruent to 3 mod 4.
1
0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 2, 1, 2, 3, 2, 2, 2, 4, 4, 3, 4, 6, 5, 5, 6, 8, 8, 7, 9, 11, 11, 10, 12, 15, 16, 15, 18, 21, 21, 21, 24, 28, 30, 29, 33, 38, 39, 40, 44, 51, 53, 54, 60, 67, 70, 72, 79, 89, 93, 96, 105, 116, 121, 126, 136, 150
OFFSET
1,15
REFERENCES
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41
G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001) See p. 235, Entry 9.4.8.
FORMULA
G.f.: x^3 / (1 - x^4) + x^8 / ((1 - x^2) * (1 - x^8)) + x^15 / ((1 - x^2) * (1 - x^4) * (1 - x^12)) + x^24 / ((1 - x^2) * (1 - x^4) * (1 - x^6) * (1 - x^16)) + ... [Ramanujan]. - Michael Somos, Jul 21 2008
2 * a(n) = A143063(n) unless n=0. - Michael Somos, Jul 09 2015
EXAMPLE
G.f. = x^3 + x^7 + x^8 + x^10 + x^11 + x^12 + x^14 + 2*x^15 + 2*x^16 + x^17 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (QHypergeometricPFQ[ {-1}, {-x^2}, x^2, -x^3] - 1) / 2, {x, 0, n}]; (* Michael Somos, Jun 25 2015 *)
PROG
(PARI) {a(n) = if( n<1, 0, polcoeff( sum(k=1, sqrtint(n+1) - 1, x^(k^2 + 2*k) / (1 - x^(4*k)) / prod(j=1, k-1, 1 - x^(2*j), 1 + O(x^(n + 1 - k^2 - 2*k)))), n))}; /* Michael Somos, Jul 21 2008 */
(PARI) {a(n) = my(A, B); if( n<1, 0, A = partitions(n); sum(k=1, length(A), if( ((B = A[k])[1])%4 == 3, prod(j=2, length(B), (B[j] > B[j-1]) && ((B[j] - B[j-1])%2 == 0)))))}; /* Michael Somos, Jul 21 2008 */
CROSSREFS
Cf. A143063.
Sequence in context: A263765 A335424 A270073 * A238325 A238885 A023566
KEYWORD
nonn
STATUS
approved