A213256 -id:A213256 - cp's OEIS frontend

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

Benoit Cloitre, Jun 24 2002

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

			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 + ...

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

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

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

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

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 *)

{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};
```

a(n)=polcoeff(prod(m=1,n,(1-x^(5*m))^5/(1-x^m +x*O(x^n))^6),n) \\ Paul D. Hanna

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

A076394 a(n) = p(11n+6)/11 where p(n) = number of partitions of n (A000041).

Original entry on oeis.org

1, 27, 338, 2835, 18566, 101955, 490253, 2121679, 8424520, 31120519, 108082568, 355805845, 1117485621, 3366123200, 9767105406, 27398618368, 74534264393, 197147918679, 508189847045, 1279140518117, 3149375120229, 7596463993261

Offset: 0

Jeff Burch, Nov 07 2002

nonn

That p(11n+6) == 0 (mod 11) is one of the congruences stated by Ramanujan. - Omar E. Pol, Jan 18 2013

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Lasse Winquist, An elementary proof of p(11m+6) == 0 (mod 11), J. Combinatorial Theory 6 1969 56--59. MR0236136 (38 #4434). - From _N. J. A. Sloane_, Jun 07 2012

Cf. A000041, A071734, A071746, A213256, A182668.

seq(combinat:-numbpart(11*n+6)/11, n=0..30); # Robert Israel, Jan 07 2015

PartitionsP[(11*Range[0,30]+6)]/11 (* Harvey P. Dale, May 28 2015 *)

a(n) = numbpart(11*n+6)/11; \\ Michel Marcus, Jan 07 2015

a(n) = A000041(A017461(n))/11 = A213256(n)/11. - Omar E. Pol, Jan 18 2013

A213261 a(n) = p(7*n + 5), where p(k) = number of partitions of k = A000041(k).

Original entry on oeis.org

7, 77, 490, 2436, 10143, 37338, 124754, 386155, 1121505, 3087735, 8118264, 20506255, 49995925, 118114304, 271248950, 607163746, 1327710076, 2841940500, 5964539504, 12292341831, 24908858009, 49686288421, 97662728555, 189334822579, 362326859895, 684957390936, 1280011042268, 2366022741845, 4328363658647, 7840656226137

Offset: 0

N. J. A. Sloane, Jun 07 2012

nonn

It is known that a(n) is divisible by 7 (see A071746).

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Ho-Hon Leung, Another Identity for Complete Bell Polynomials based on Ramanujan's Congruences, arXiv:1802.08443 [math.CO], 2018.
Ho-Hon Leung, Another Identity for Complete Bell Polynomials based on Ramanujan's Congruences, J. Integer Seq. 21 (2018), Article 18.6.4.
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

Cf. A000041, A017041, A071734, A071746, A076394, A213256, A213260, A213261, A327582, A327714, A327770.

Table[PartitionsP[7 n + 5], {n, 0, 29}] (* Jean-François Alcover, Nov 12 2018 *)

a(n) = numbpart(7*n+5); \\ Michel Marcus, Jan 07 2015

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

a(n) = 7 * A071746(n). - Joerg Arndt, Nov 06 2016

A213260 p(5n+4) where p(k) = number of partitions of k = A000041(k).

Original entry on oeis.org

5, 30, 135, 490, 1575, 4565, 12310, 31185, 75175, 173525, 386155, 831820, 1741630, 3554345, 7089500, 13848650, 26543660, 49995925, 92669720, 169229875, 304801365, 541946240, 952050665, 1653668665, 2841940500, 4835271870, 8149040695, 13610949895, 22540654445, 37027355200, 60356673280, 97662728555, 156919475295

Offset: 0

N. J. A. Sloane, Jun 07 2012

nonn

It is known that a(n) is divisible by 5 (see A071734).

Seiichi Manyama, Table of n, a(n) for n = 0..1000
James Grime and Brady Haran, Partitions, Numberphile video (2016).
Lasse Winquist, An elementary proof of p(11m+6) == 0 (mod 11), J. Combinatorial Theory 6 1969 56--59. MR0236136 (38 #4434). - From _N. J. A. Sloane_, Jun 07 2012

Cf. A000041, A071734, A213256, A076394.

Table[PartitionsP[5n+4],{n,0,40}] (* Harvey P. Dale, May 30 2013 *)

a(n) = numbpart(5*n+4); \\ Michel Marcus, Jan 07 2015

from sympy.functions import partition
def a(n): return partition(5*n+4)
print([a(n) for n in range(33)]) # Michael S. Branicky, May 30 2021

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

A220503 spt(13n+6) where spt(n) = A092269(n).

Original entry on oeis.org

26, 1820, 39546, 494702, 4474756, 32347380, 198063060, 1065041120, 5155845968, 22871059718, 94204920680, 363981624370, 1329826483453, 4624153352104, 15385030362884, 49194117590072, 151738308808580, 452922550115880, 1311857021146256

Offset: 0

Omar E. Pol, Jan 18 2013

nonn

a(n) is divisible by 13 (see A220513).

G. E. Andrews, The number of smallest parts in the partitions of n

Cf. A092269, A186113, A213256, A220485, A220502, A220513.

a(n) = A092269(A186113(n)).

A023922 Theta series of A*_10 lattice.

Original entry on oeis.org

1, 0, 0, 0, 0, 22, 0, 0, 0, 110, 0, 110, 330, 0, 660, 924, 990, 0, 0, 0, 2662, 0, 1980, 4840, 0, 6534, 7260, 9460, 0, 0, 0, 15840, 0, 10230, 21780, 0, 27830, 32670, 33660, 0, 0, 0, 50820, 0, 30140, 71236, 0, 84700, 87120, 99000, 0, 0, 0, 136840, 0

Offset: 0

N. J. A. Sloane

nonn

			G.f. = 1 + 22*q^5 + 110*q^9 + 110*q^11 + 330*q^12 + ... - _Sean A. Irvine_, Jun 16 2019

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 114.

G. Nebe and N. J. A. Sloane, Home page for this lattice
K. Ono, On the Circular Summation of the Eleventh Powers of Ramanujan's Theta Function, Journal of Number Theory, Volume 76, Issue 1, May 1999, Pages 62-65.

G.f.: eta^11(q)/eta(q^11) + 800*q^5*eta^11(q^11)/eta(11) - 9*P11(q)*eta^11(q) + (308/3)*A(q) + (22/3)*B(q), where P11(q) is the g.f. for A213256, A(q) is the g.f. for A065103, B(q) is the g.f. for A065099, and eta is the Dedekind eta function [from Ono]. - Sean A. Irvine, Jun 16 2019

More terms from Sean A. Irvine, Jun 16 2019

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

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A076394 a(n) = p(11n+6)/11 where p(n) = number of partitions of n (A000041).

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A213261 a(n) = p(7*n + 5), where p(k) = number of partitions of k = A000041(k).

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A213260 p(5n+4) where p(k) = number of partitions of k = A000041(k).

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A220503 spt(13n+6) where spt(n) = A092269(n).

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A023922 Theta series of A*_10 lattice.

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