cp's OEIS Frontend

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

%I A027348 #15 Jul 09 2015 15:14:34
%S A027348 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,
%T A027348 11,11,10,12,15,16,15,18,21,21,21,24,28,30,29,33,38,39,40,44,51,53,54,
%U A027348 60,67,70,72,79,89,93,96,105,116,121,126,136,150
%N A027348 Number of partitions of n into distinct odd parts, the least being congruent to 3 mod 4.
%D A027348 Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41
%D A027348 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.
%F A027348 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
%F A027348 2 * a(n) = A143063(n) unless n=0. - _Michael Somos_, Jul 09 2015
%e A027348 G.f. = x^3 + x^7 + x^8 + x^10 + x^11 + x^12 + x^14 + 2*x^15 + 2*x^16 + x^17 + ...
%t A027348 a[ n_] := SeriesCoefficient[ (QHypergeometricPFQ[ {-1}, {-x^2}, x^2, -x^3] - 1) / 2, {x, 0, n}]; (* _Michael Somos_, Jun 25 2015 *)
%o A027348 (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 */
%o A027348 (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 */
%Y A027348 Cf. A143063.
%K A027348 nonn
%O A027348 1,15
%A A027348 _Clark Kimberling_