cp's OEIS Frontend

A071734 a(n) = p(5n+4)/5 where p(k) denotes the k-th partition number.

%I A071734 #69 Aug 11 2025 06:08:52
%S A071734 1,6,27,98,315,913,2462,6237,15035,34705,77231,166364,348326,710869,
%T A071734 1417900,2769730,5308732,9999185,18533944,33845975,60960273,108389248,
%U A071734 190410133,330733733,568388100,967054374,1629808139,2722189979
%N A071734 a(n) = p(5n+4)/5 where p(k) denotes the k-th partition number.
%C A071734 One of the congruences related to the partition numbers stated by Ramanujan.
%C A071734 Also the coefficients in the expansion of C^5/B^6, in Watson's notation (p. 105). The connection to the partition function is in equation (3.31) with right side 5C^5/B^6 where B = x * f(-x^24), C = x^5 * f(-x^120) where f() is a Ramanujan theta function. Alternatively B = eta(q^24), C = eta(q^120). - _Michael Somos_, Jan 06 2015
%D A071734 Berndt and Rankin, "Ramanujan: letters and commentaries", AMS-LMS, History of mathematics, vol. 9, pp. 192-193
%D A071734 G. H. Hardy, Ramanujan, Cambridge Univ. Press, 1940. - From _N. J. A. Sloane_, Jun 07 2012
%H A071734 Seiichi Manyama, <a href="/A071734/b071734.txt">Table of n, a(n) for n = 0..1000</a>
%H A071734 S. Bouroubi and N. Benyahia Tani, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Bouroubi/bouroubi25.html">A new identity for complete Bell polynomials based on a formula of Ramanujan</a>, J. Integer Seq. 12 (2009), 09.3.5.
%H A071734 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 A071734 M. D. Hirschhorn, <a href="http://web.maths.unsw.edu.au/~mikeh/webpapers/paper60.pdf">Another Shorter Proof of Ramanujan's Mod 5 Partition Congruence, and More</a>, Amer. Math. Monthly 106(6) (1999), 580-583.
%H A071734 M. Savic, <a href="http://www.cs.bsu.edu/homepages/fischer/Journal/01-01/savic.pdf">The Partition Function and Ramanujan's 5k+4 Congruence</a>, Mathematics Exchange 1(1) (2003), 2-4.
%H A071734 G. N. Watson, <a href="https://gdz.sub.uni-goettingen.de/id/PPN243919689_0179">Ramanujans Vermutung über Zerfällungszahlen</a>, J. Reine Angew. Math. (Crelle) 179 (1938), 97-128.
%H A071734 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 A071734 a(n) = (1/5)*A000041(5n+4).
%F A071734 G.f.: Product_{n>=1} (1 - x^(5*n))^5/(1 - x^n)^6 due to Ramanujan's identity. - _Paul D. Hanna_, May 22 2011
%F A071734 a(n) = A000041(A016897(n))/5 = A213260(n)/5. - _Omar E. Pol_, Jan 18 2013
%F A071734 Euler transform of period 5 sequence [ 6, 6, 6, 6, 1, ...]. - _Michael Somos_, Jan 07 2015
%F A071734 Expansion of q^(-19/24) * eta(q^5)^5 / eta(q)^6 in powers of q. - _Michael Somos_, Jan 07 2015
%F A071734 a(n) ~ exp(Pi*sqrt(10*n/3)) / (100*sqrt(3)*n). - _Vaclav Kotesovec_, Nov 28 2016
%e A071734 G.f. = 1 + 6*x + 27*x^2 + 98*x^3 + 315*x^4 + 913*x^5 + 2462*x^6 + ...
%e A071734 G.f. = q^19 + 6*q^43 + 27*q^67 + 98*q^91 + 315*q^115 + 913*q^139 + ...
%p A071734 with(combinat):
%p A071734 a:= n-> numbpart(5*n+4)/5:
%p A071734 seq(a(n), n=0..40);  # _Alois P. Heinz_, Jan 07 2015
%t A071734 a[ n_] := PartitionsP[ 5 n + 4] / 5; (* _Michael Somos_, Jan 07 2015 *)
%t A071734 a[ n_] := SeriesCoefficient[ 1 / QPochhammer[ x], {x, 0, 5 n + 4}] / 5; (* _Michael Somos_, Jan 07 2015 *)
%t A071734 nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))^5/(1 - x^k)^6, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 28 2016 *)
%o A071734 (PARI) {a(n) = if( n<0, 0, polcoeff( 1 / eta(x + O(x^(5*n + 5))), 5*n + 4) / 5)};
%o A071734 (PARI) {a(n) = numbpart(5*n + 4) / 5};
%o A071734 (PARI) a(n)=polcoeff(prod(m=1,n,(1-x^(5*m))^5/(1-x^m +x*O(x^n))^6),n) \\ _Paul D. Hanna_
%Y A071734 Cf. A000041, A016897, A071746, A076394, A213256, A213260.
%K A071734 easy,nonn
%O A071734 0,2
%A A071734 _Benoit Cloitre_, Jun 24 2002