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

Author

Clark Kimberling

Keywords

• nonn

Examples

			G.f. = x^3 + x^7 + x^8 + x^10 + x^11 + x^12 + x^14 + 2*x^15 + 2*x^16 + x^17 + ...
		

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• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41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

Crossrefs

Cf. A143063.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ (QHypergeometricPFQ[ {-1}, {-x^2}, x^2, -x^3] - 1) / 2, {x, 0, n}]; (* Michael Somos, Jun 25 2015 *)

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 */

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