A071746 - OEIS

%I #37 Sep 08 2022 08:45:06

%S 1,11,70,348,1449,5334,17822,55165,160215,441105,1159752,2929465,

%T 7142275,16873472,38749850,86737678,189672868,405991500,852077072,

%U 1756048833,3558408287,7098041203,13951818365,27047831797,51760979985

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

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

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

%D G. H. Hardy, Ramanujan, Cambridge Univ. Press, 1940. - From _N. J. A. Sloane_, Jun 07 2012

%H Seiichi Manyama, <a href="/A071746/b071746.txt">Table of n, a(n) for n = 0..1000</a>

%H J. L. Drost, <a href="http://www.jstor.org/stable/2974479">A Shorter Proof of the Ramanujan Congruence Modulo 5</a>, Amer. Math. Monthly 104(10) (1997), 963-964.

%H Lasse Winquist, <a href="http://dx.doi.org/10.1016/S0021-9800(69)80105-5">An elementary proof of p(11m+6) == 0 (mod 11)</a>, J. Combinatorial Theory 6(1) (1969), 56-59. MR0236136 (38 #4434). - From _N. J. A. Sloane_, Jun 07 2012

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

%F a(n) = A000041(A017041(n))/7 = A213261(n)/7. - _Omar E. Pol_, Jan 18 2013

%t Table[PartitionsP[7n+5]/7, {n, 0, 24}] (* _Jean-François Alcover_, Nov 30 2015 *)

%o (PARI) a(n)=if(n<0, 0, n=7*n+5; polcoeff(1/eta(x+x*O(x^n)),n)/7)

%o (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 */

%o (PARI) a(n) = numbpart(7*n+5)/7; \\ _Michel Marcus_, Nov 30 2015

%o (Magma) a:= func< n | NumberOfPartitions((7*n+5)) div 7 >; [ a(n) : n in [0..30]]; // _Vincenzo Librandi_, Nov 30 2015

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

%K easy,nonn

%O 0,2

%A _Benoit Cloitre_, Jun 24 2002