A071746 -id:A071746 - 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

A160527 Coefficients in the expansion of C^3/B^4, in Watson's notation of page 118.

Original entry on oeis.org

1, 4, 14, 40, 105, 252, 574, 1237, 2568, 5138, 9988, 18893, 34937, 63238, 112370, 196244, 337477, 572024, 956956, 1581321, 2583637, 4176495, 6684820, 10599939, 16661401, 25972485, 40171474, 61672695, 94017765, 142368024, 214211760, 320350725, 476299978

Offset: 0

N. J. A. Sloane, Nov 13 2009

nonn

			G.f. = 1 + 4*x + 14*x^2 + 40*x^3 + 105*x^4 + 252*x^5 + 574*x^6 + ...
G.f. = q^17 + 4*q^41 + 14*q^65 + 40*q^89 + 105*q^113 + 252*q^137 + 574*q^161 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128.

Cf. A071746, A213261.

nmax = 50; CoefficientList[Series[Product[(1 - x^(7*k))^3 /(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2017 *)

See Maple code in A160525 for formula.
G.f.: Product_{n >= 1} (1 - x^(7*n))^3/(1 - x^n)^4. - Seiichi Manyama, Nov 06 2016
a(n) ~ exp(Pi*sqrt(50*n/21)) * 5 / (196*sqrt(3)*n). - Vaclav Kotesovec, Nov 10 2017

A327770 a(n) = (23 * 7^(2*n) + 1)/24. Sequence related to the properties of the partition function A000041 modulo a power of 7.

Original entry on oeis.org

1, 47, 2301, 112747, 5524601, 270705447, 13264566901, 649963778147, 31848225129201, 1560563031330847, 76467588535211501, 3746911838225363547, 183598680073042813801, 8996335323579097876247, 440820430855375795936101, 21600201111913414000868947

Offset: 0

Petros Hadjicostas, Sep 24 2019

If p(n) = A000041(n) is the partition function, Watson (1938) proved that p(7^(2*m)*n + a(m)) == 0 mod 7^(m+1) for n >= 0 and m >= 1. (Obviously, this is not always true for m = 0).
For m=1 and n=0, p(7^(2*1)*0 + a(1)) = p(47) = 7^(1+1) * 2546.
For m=1 and n=1, p(7^(2*1)*1 + a(1)) = p(96) = 7^(1+1) * 2410496.
For m=1 and n=2, p(7^(2*1)*2 + a(1)) = p(145) = 7^(1+1) * 508344041.
For m=2 and n=0, p(7^(2*2)*0 + a(2)) = p(2301) = 7^(2+1) * 49629361905981812695622866669844910256876089360.
Essentially the same as A052463. - R. J. Mathar, Oct 08 2019

Colin Barker, Table of n, a(n) for n = 0..500
G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128; see pp. 118 and 124.
Eric Weisstein's World of Mathematics, Partition Function P Congruences.
Wikipedia, G. N. Watson.
Index entries for linear recurrences with constant coefficients, signature (50,-49).

Cf. A052463, A071746, A213261, A327714, A327582.

CoefficientList[Series[(1 - 3 x)/((1 - x) (1 - 49 x)), {x, 0, 15}], x] (* Michael De Vlieger, Sep 27 2019 *)
LinearRecurrence[{50,-49},{1,47},20] (* Harvey P. Dale, Mar 09 2023 *)

a(n) = (23 * 7^(2*n) + 1)/24; \\ Michel Marcus, Sep 25 2019

Vec((1 - 3*x) / ((1 - x)*(1 - 49*x)) + O(x^20)) \\ Colin Barker, Sep 25 2019

From Colin Barker, Sep 25 2019: (Start)
G.f.: (1 - 3*x) / ((1 - x)*(1 - 49*x)).
a(n) = 50*a(n-1) - 49*a(n-2) for n>1.
(End)

A327582 a(n) = (17 * 7^(2*n+1) + 1)/24. Sequence related to the properties of the partition function A000041 modulo a power of 7.

Original entry on oeis.org

5, 243, 11905, 583343, 28583805, 1400606443, 68629715705, 3362856069543, 164779947407605, 8074217422972643, 395636653725659505, 19386196032557315743, 949923605595308471405, 46546256674170115098843, 2280766577034335639843305, 111757562274682446352321943

Offset: 0

Petros Hadjicostas, Sep 23 2019

If p(n) = A000041(n) is the partition function, Watson (1938) proved that p(7^(2*m+1)*n + a(m)) == 0 mod 7^(m+1) for n >= 0 and m >= 1.

It is well-known that this result is true even for m = 0 (cf. A071746 and the references there).

			For m=1 and n=0, p(7^(2*1+1)*0 + a(1)) = p(243) = 133978259344888 = 7^2 * 2734250190712.
For m=1 and n=1, p(7^(2*1+1)*1 + a(1)) = p(586) = 224282898599046831034631 = 7^2 * 4577202012225445531319.

Colin Barker, Table of n, a(n) for n = 0..500
G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128; see pp. 118 and 124.
Eric Weisstein's World of Mathematics, Partition Function P Congruences.
Wikipedia, G. N. Watson.
Index entries for linear recurrences with constant coefficients, signature (50,-49).

Cf. A000041, A071746, A110375, A213261, A327714, A327770.

a(n) = (17 * 7^(2*n+1) + 1)/24; \\ Michel Marcus, Sep 25 2019

Vec((5 - 7*x) / ((1 - x)*(1 - 49*x)) + O(x^15)) \\ Colin Barker, Sep 27 2019

From Colin Barker, Sep 27 2019: (Start)
G.f.: (5 - 7*x) / ((1 - x)*(1 - 49*x)).
a(n) = 50*a(n-1) - 49*a(n-2) for n>1.
(End)

A220507 a(n) = spt(7n+5)/7 where spt(n) = A092269(n).

Original entry on oeis.org

2, 34, 260, 1498, 6956, 28024, 100953, 333680, 1026540, 2976024, 8197962, 21608760, 54788100, 134217717, 318816426, 736549424, 1659169712, 3652248590, 7870890952, 16633964444, 34522173765, 70450341042, 141526909340, 280158178412

Offset: 0

Omar E. Pol, Jan 18 2013

nonn

That spt(7n+5) == 0 (mod 7) is one of the congruences stated by George E. Andrews. See theorem 2 in the Andrews' paper. See also A220505 and A220513.

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. A017041, A071746, A092269, A220502, A220505, A220513.

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[7 n + 5]/7;
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jan 30 2019, after Alois P. Heinz in A092269 *)

a(n) = A092269(A017041(n))/7 = A220502(n)/7.

A277958 Expansion of Product_{n>=1} (1 - x^(7*n))^7/(1 - x^n)^8 in powers of x.

Original entry on oeis.org

1, 8, 44, 192, 726, 2464, 7704, 22521, 62281, 164252, 415796, 1015334, 2401462, 5519640, 12363062, 27047913, 57917068, 121588588, 250638216, 507974950, 1013409244, 1992161935, 3862461694, 7392045512, 13975011909, 26116935550, 48277368020, 88320521108, 159993054081

Offset: 0

Seiichi Manyama, Nov 06 2016

nonn

			G.f.: 1 + 8*x + 44*x^2 + 192*x^3 + 726*x^4 + 2464*x^5 + 7704*x^6 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Wikipedia, Ramanujan's congruences

Cf. A071746, A160527, A213261.

nmax = 20; CoefficientList[Series[Product[(1 - x^(7*k))^7 /(1 - x^k)^8 , {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2017 *)

G.f.: Product_{n>=1} (1 - x^(7*n))^7/(1 - x^n)^8.
A213261(n) = 7*A160527(n) + 49*a(n - 1) for n > 0 due to Ramanujan's congruences.
a(n) ~ exp(Pi*sqrt(98*n/21)) / (1372*sqrt(3)*n). - Vaclav Kotesovec, Nov 10 2017

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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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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A160527 Coefficients in the expansion of C^3/B^4, in Watson's notation of page 118.

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A327770 a(n) = (23 * 7^(2*n) + 1)/24. Sequence related to the properties of the partition function A000041 modulo a power of 7.

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A327582 a(n) = (17 * 7^(2*n+1) + 1)/24. Sequence related to the properties of the partition function A000041 modulo a power of 7.

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A220507 a(n) = spt(7n+5)/7 where spt(n) = A092269(n).

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