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

1, 6, 27, 98, 315, 913, 2462, 6237, 15035, 34705, 77231, 166364, 348326, 710869, 1417900, 2769730, 5308732, 9999185, 18533944, 33845975, 60960273, 108389248, 190410133, 330733733, 568388100, 967054374, 1629808139, 2722189979

Offset: 0

Author

Benoit Cloitre, Jun 24 2002

Keywords

• easy
• nonn

Comments

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

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

Examples

			G.f. = 1 + 6*x + 27*x^2 + 98*x^3 + 315*x^4 + 913*x^5 + 2462*x^6 + ...
G.f. = q^19 + 6*q^43 + 27*q^67 + 98*q^91 + 315*q^115 + 913*q^139 + ...
		

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
• S. Bouroubi and N. Benyahia Tani, A new identity for complete Bell polynomials based on a formula of Ramanujan, J. Integer Seq. 12 (2009), 09.3.5.
• J. L. Drost, A Shorter Proof of the Ramanujan Congruence Modulo 5, Amer. Math. Monthly 104(10) (1997), 963-964.
• M. D. Hirschhorn, Another Shorter Proof of Ramanujan's Mod 5 Partition Congruence, and More, Amer. Math. Monthly 106(6) (1999), 580-583.
• M. Savic, The Partition Function and Ramanujan's 5k+4 Congruence, Mathematics Exchange 1(1) (2003), 2-4.
• G. N. Watson, Ramanujans Vermutung über Zerfällungszahlen, J. Reine Angew. Math. (Crelle) 179 (1938), 97-128.
• 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

Crossrefs

Cf. A000041, A016897, A071746, A076394, A213256, A213260.

Programs

Maple

with(combinat):
a:= n-> numbpart(5*n+4)/5:
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 07 2015

Mathematica

a[ n_] := PartitionsP[ 5 n + 4] / 5; (* Michael Somos, Jan 07 2015 *)
a[ n_] := SeriesCoefficient[ 1 / QPochhammer[ x], {x, 0, 5 n + 4}] / 5; (* Michael Somos, Jan 07 2015 *)
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 *)

PARI

{a(n) = if( n<0, 0, polcoeff( 1 / eta(x + O(x^(5*n + 5))), 5*n + 4) / 5)};

PARI

{a(n) = numbpart(5*n + 4) / 5};

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

Formula

a(n) = (1/5)*A000041(5n+4).

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

a(n) = A000041(A016897(n))/5 = A213260(n)/5. - Omar E. Pol, Jan 18 2013

Euler transform of period 5 sequence [ 6, 6, 6, 6, 1, ...]. - Michael Somos, Jan 07 2015

Expansion of q^(-19/24) * eta(q^5)^5 / eta(q)^6 in powers of q. - Michael Somos, Jan 07 2015

a(n) ~ exp(Pi*sqrt(10*n/3)) / (100*sqrt(3)*n). - Vaclav Kotesovec, Nov 28 2016