A071746 a(n) = p(7n+5)/7 where p(k) denotes the k-th partition number.

1, 11, 70, 348, 1449, 5334, 17822, 55165, 160215, 441105, 1159752, 2929465, 7142275, 16873472, 38749850, 86737678, 189672868, 405991500, 852077072, 1756048833, 3558408287, 7098041203, 13951818365, 27047831797, 51760979985

Offset

0,2

Comments

One of the congruences related to the partition numbers stated by Ramanujan.

References

Berndt and Rankin, "Ramanujan: letters and commentaries", AMS-LMS, History of mathematics, vol. 9, pp. 192-193.

G. H. Hardy, Ramanujan, Cambridge Univ. Press, 1940. - From N. J. A. Sloane, Jun 07 2012

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

J. L. Drost, A Shorter Proof of the Ramanujan Congruence Modulo 5, Amer. Math. Monthly 104(10) (1997), 963-964.

Lasse Winquist, An elementary proof of p(11m+6) == 0 (mod 11), J. Combinatorial Theory 6(1) (1969), 56-59. MR0236136 (38 #4434). - From N. J. A. Sloane, Jun 07 2012

Formula

a(n) = (1/7)*A000041(7n+5).

a(n) = A000041(A017041(n))/7 = A213261(n)/7. - Omar E. Pol, Jan 18 2013

Mathematica

Table[PartitionsP[7n+5]/7, {n, 0, 24}] (* Jean-François Alcover, Nov 30 2015 *)

Prog

(PARI) a(n)=if(n<0, 0, n=7*n+5; polcoeff(1/eta(x+x*O(x^n)), n)/7)
(PARI) {a(n)=local(A, B); if(n<0, 0, A=x*O(x^n); B=eta(x^7+A); A=eta(x+A); polcoeff( B^3/A^4 +x*7*B^7/A^8, n))} /* Michael Somos, Jan 01 2006 */
(PARI) a(n) = numbpart(7*n+5)/7; \\ Michel Marcus, Nov 30 2015
(Magma) a:= func< n | NumberOfPartitions((7*n+5)) div 7 >; [ a(n) : n in [0..30]]; // Vincenzo Librandi, Nov 30 2015

Crossrefs

Cf. A000041, A017041, A071734, A076394, A213261, A327582, A327714, A327770.

Sequence in context: A211050 A295074 A173200 * A205812 A162568 A255205

Adjacent sequences: A071743 A071744 A071745 * A071747 A071748 A071749

Keyword

easy,nonn

Author

Benoit Cloitre, Jun 24 2002

Status

approved