A027348 Number of partitions of n into distinct odd parts, the least being congruent to 3 mod 4.

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41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.

Links

Table of n, a(n) for n=1..71.

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

Adjacent sequences: A027345 A027346 A027347 * A027349 A027350 A027351

Keyword

nonn

Author

Clark Kimberling

Status

approved