A076394 -id:A076394 - 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

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

Original entry on oeis.org

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

Benoit Cloitre, Jun 24 2002

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

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
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

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

a:= func< n | NumberOfPartitions((7*n+5)) div 7 >; [ a(n) : n in [0..30]]; // Vincenzo Librandi, Nov 30 2015

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

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

{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 */

a(n) = numbpart(7*n+5)/7; \\ Michel Marcus, Nov 30 2015

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

a(n) = A000041(A017041(n))/7 = A213261(n)/7. - 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

A213256 p(11n+6) where p(k) = number of partitions of k = A000041(k).

Original entry on oeis.org

11, 297, 3718, 31185, 204226, 1121505, 5392783, 23338469, 92669720, 342325709, 1188908248, 3913864295, 12292341831, 37027355200, 107438159466, 301384802048, 819876908323, 2168627105469, 5590088317495, 14070545699287, 34643126322519, 83561103925871, 197726516681672, 459545750448675, 1050197489931117

Offset: 0

N. J. A. Sloane, Jun 07 2012

nonn

It is known that a(n) is divisible by 11 (see A076394).

Seiichi Manyama, Table of n, a(n) for n = 0..1000
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.
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, A076394.

PartitionsP[11Range[0,30]+6] (* Paolo Xausa, Nov 08 2023 *)

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

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

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

A220513 a(n) = spt(13n+6)/13 where spt(n) = A092269(n).

Original entry on oeis.org

2, 140, 3042, 38054, 344212, 2488260, 15235620, 81926240, 396603536, 1759312286, 7246532360, 27998586490, 102294344881, 355704104008, 1183463874068, 3784162891544, 11672177600660, 34840196162760, 100912078549712, 284295561826160

Offset: 0

Omar E. Pol, Jan 18 2013

nonn

That spt(13n+6) == 0 (mod 13) is one of the congruences stated by George E. Andrews. See theorem 2 in the Andrews' paper. See also A220505 and A220507.

G. E. Andrews, The number of smallest parts in the partitions of n
G. E. Andrews, F. G. Garvan, and J. Liang, Combinatorial interpretation of congruences for the spt-function
K. C. Garrett, C. McEachern, T. Frederick, O. Hall-Holt, Fast computation of Andrews' smallest part statistic and conjectured congruences, Discrete Applied Mathematics, 159 (2011), 1377-1380.
F. G. Garvan, Congruences for Andrews' smallest parts partition function and new congruences for Dyson's rank
F. G. Garvan, Congruences for Andrews' spt-function modulo powers of 5, 7 and 13
F. G. Garvan, Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences, arXiv:1011.1957 [math.NT], 2010.
K. Ono, Congruences for the Andrews spt-function

Cf. A076394, A092269, A186113, A220503, A220505, A220507.

b[n_, i_] := b[n, i] = If[n==0 || i==1, n, {q, r} = QuotientRemainder[n, i]; If[r == 0, q, 0] + Sum[b[n - i*j, i - 1], {j, 0, n/i}]];
spt[n_] := b[n, n];
a[n_] := spt[13 n + 6]/13;
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Jan 30 2019, after Alois P. Heinz in A092269 *)

a(n) = A092269(A186113(n))/13 = A220503(n)/13.

A182668 The n-th Fourier coefficient divided by 11 of L_1(tau) defined by A. O. L. Atkin in 1967.

Original entry on oeis.org

1, 27, 338, 2835, 18566, 101955, 490253, 2121679, 8424520, 31120519, 108082568, 355805844, 1117485594, 3366122862, 9767102571, 27398599802, 74534162438, 197147428426, 508187725366, 1279132093597, 3149343999710, 7596355910693, 17974782074306, 41775768918777

Offset: 1

Michael Somos, Dec 24 2012

nonn

Atkin (1967) on page 22, equation (30), defines phi(tau) = eta(121*tau) / eta(tau), a modular function which satisfies phi(-1/(121*t)) = 11^(-1)/phi(t), where q = exp(2*Pi*i*t). On page 23, equation (33), he defines L_1(tau) = U phi(tau), where U is a Hecke operator so that the n-th Fourier coefficient of L_1 is the 11*n-th Fourier coefficient of phi. On page 26, he finds that L_1(tau) = 11*g_2(tau) + 2*11^2*g_3(tau) + 11^3*g_4(tau) + 11^4*g_5(tau), where g_2, g_3, g_4, g_5 are functions he previously defined. The n-th Fourier coefficient of L_1 is 11*a(n).
First differs from A076394 at a(12). - Omar E. Pol, Dec 24 2012
The sequence of coefficients of the q-expansion of phi(tau) coincides with the partition function A000041 for the first 120 terms. - N. J. A. Sloane, Dec 24 2012

			x + 27*x^2 + 338*x^3 + 2835*x^4 + 18566*x^5 + 101955*x^6 + 490253*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
A. O. L. Atkin, Proof of a conjecture of Ramanujan, Glasgow Math. J., 8 (1967), 14-32.

Cf. A000041, A076394.

eta[q_] := q^(1/24)*QPochhammer[q]; CoefficientList[Series[ eta[q^121]/ eta[q]/11, {q, 0, 300}], q][[1 ;; -1 ;; 11]] (* G. C. Greubel, Aug 10 2018 *)

{a(n) = local(A); if( n<1, 0, n = 11*n - 5; A = x * O(x^n); polcoeff( eta(x^121 + A) / eta(x + A), n) / 11)}

A333435 Partition numbers A000041(k*x_n + y_n) are known to be divisible by prime(n); sequence gives the list of y_n.

Original entry on oeis.org

4, 5, 6, 237, 2623, 815655

Offset: 3

Frank Ellermann, Mar 21 2020

Grime notes that Ramanujan's pattern for a(3), a(4), a(5) or prime(3), prime(4), prime(5) cannot be directly extended to prime(6) = 13, and shows solutions for 13, 17, 19.

			All {partition( 5k+4)} are divisible by prime(3) = 5, so a(3) = 4.
All {partition( 7k+5)} are divisible by prime(4) = 7, so a(4) = 5.
All {partition(11k+6)} are divisible by prime(5) = 11, so a(5) = 6.

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).

Cf. A333436 (y_n), A000040 (primes), A000041 (partitions).
Cf. A071734 (p(5k+4)/5), A071746 (p(7k+5)/7), A076394 (p(11k+6)/11).
Cf. A213260 (p(5k+4)).

A333436 Partition numbers A000041(k*x_n + y_n) are known to be divisible by prime(n); sequence gives the list of x_n.

Original entry on oeis.org

5, 7, 11, 17303, 206839, 1977147619

Offset: 3

Frank Ellermann, Mar 21 2020

Grime notes that Ramanujan's pattern for a(3), a(4), a(5) or prime(3), prime(4), prime(5) cannot be directly extended to prime(6) = 13, and shows solutions for 13, 17, 19.

			All {partition( 5k+4)} are divisible by prime(3) = 5, so a(3) = 5.
All {partition( 7k+5)} are divisible by prime(4) = 7, so a(4) = 7.
All {partition(11k+6)} are divisible by prime(5) = 11, so a(5) = 11.

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).

Cf. A333435 (x_n), A000040 (primes), A000041 (partitions).
Cf. A071734 (p(5k+4)/5), A071746 (p(7k+5)/7), A076394 (p(11k+6)/11).
Cf. A213260 (p(5k+4)).

cp's OEIS Frontend

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

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

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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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A213256 p(11n+6) 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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A220513 a(n) = spt(13n+6)/13 where spt(n) = A092269(n).

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A182668 The n-th Fourier coefficient divided by 11 of L_1(tau) defined by A. O. L. Atkin in 1967.

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