A071734 -id:A071734 - cp's OEIS frontend

A160507 Duplicate of A071734.

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, 4508130889, 7405471040

Offset: 0

dead

A160461 Coefficients in the expansion of C/B^2, in Watson's notation of page 106.

Original entry on oeis.org

1, 2, 5, 10, 20, 35, 63, 105, 175, 280, 444, 685, 1050, 1575, 2345, 3439, 5005, 7195, 10275, 14525, 20405, 28428, 39375, 54150, 74080, 100715, 136265, 183365, 245645, 327485, 434810, 574790, 756965, 992950, 1297940, 1690500, 2194642, 2839695, 3663225, 4711160

Offset: 0

N. J. A. Sloane, Nov 13 2009

			x^3+2*x^27+5*x^51+10*x^75+20*x^99+35*x^123+63*x^147+...

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

Cf. Product_{n>=1} (1 - x^(5*n))^k/(1 - x^n)^(k + 1): this sequence (k=1), A160462 (k=2), A160463 (k=3), A160506 (k=4), A071734 (k=5), A160460 (k=6), A160521 (k=7), A278555 (k=12), A278556 (k=18), A278557 (k=24), A278558 (k=30).

Cf. A000041, A015128, A278690, A298311.

nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 26 2016 *)

See Maple code in A160458 for formula.
a(n) ~ sqrt(3)*exp(sqrt(6*n/5)*Pi)/(20*n). - Vaclav Kotesovec, Nov 26 2016
G.f.: 1/Product_{n > = 1} ( 1 - x^(n/gcd(n,k)) ) for k = 5. Cf. A000041 (k = 1), A015128 (k = 2), A278690 (k = 3) and A298311 (k = 4). - Peter Bala, Nov 17 2020

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

A160460 Coefficients in the expansion of C^6 / B^7, in Watson's notation of page 106.

Original entry on oeis.org

1, 7, 35, 140, 490, 1541, 4480, 12195, 31465, 77525, 183626, 420077, 932030, 2011905, 4237130, 8725671, 17605602, 34861815, 67848095, 129946805, 245203642, 456303872, 838178470, 1520969100, 2728472695, 4841909821, 8504898720, 14794863270, 25500965320

Offset: 0

N. J. A. Sloane, Nov 13 2009

nonn

			x^23 + 7*x^47 + 35*x^71 + 140*x^95 + 490*x^119 + 1541*x^143 + ...

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

Cf. Product_{n>=1} (1 - x^(5*n))^k/(1 - x^n)^(k + 1): A160461 (k=1), A160462 (k=2), A160463 (k=3), A160506 (k=4), A071734 (k=5), this sequence (k=6), A160521 (k=7), A278555 (k=12), A278556 (k=18), A278557 (k=24), A278558 (k=30).

nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))^6/(1 - x^k)^7, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2016 *)

See Maple code in A160458 for formula.

a(n) ~ sqrt(29/15) * exp(Pi*sqrt(58*n/15)) / (500*n). - Vaclav Kotesovec, Nov 28 2016

A278556 Expansion of Product_{n>=1} (1 - x^(5*n))^18/(1 - x^n)^19 in powers of x.

Original entry on oeis.org

1, 19, 209, 1710, 11495, 66862, 347339, 1645875, 7221520, 29668595, 115116233, 424720338, 1498263563, 5076482415, 16583497160, 52399330389, 160586833362, 478482249548, 1388989067820, 3935549005725, 10901608510397, 29565343541110, 78604103339462

Offset: 0

Seiichi Manyama, Nov 23 2016

nonn

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

Cf. Product_{n>=1} (1 - x^(5*n))^k/(1 - x^n)^(k + 1): A160461 (k=1), A160462 (k=2), A160463 (k=3), A160506 (k=4), A071734 (k=5), A160460 (k=6), A160521 (k=7), A278555 (k=12), this sequence (k=18), A278557 (k=24), A278558 (k=30).

CoefficientList[ Series[ Product[(1 - x^(5n))^18/(1 - x^n)^19, {n, 22}], {x, 0, 22}], x] (* Robert G. Wilson v, Nov 24 2016 *)

G.f.: Product_{n>=1} (1 - x^(5*n))^18/(1 - x^n)^19.
A278559(n) = 5^2*63*A160460(n) + 5^5*52*A278555(n-1) + 5^7*63*a(n-2) + 5^10*6*A278557(n-3) + 5^12*A278558(n-4) for n >= 4.
a(n) ~ sqrt(77/15) * exp(Pi*sqrt(154*n/15)) / (7812500*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

A160462 Coefficients in the expansion of C^2/B^3, in Watson's notation of page 106.

Original entry on oeis.org

1, 3, 9, 22, 51, 106, 215, 411, 766, 1377, 2423, 4154, 7001, 11567, 18834, 30195, 47809, 74735, 115585, 176847, 268064, 402598, 599695, 886116, 1299808, 1893115, 2739248, 3938491, 5629407, 8000431, 11309295, 15904003, 22256183, 30998479, 42981170, 59337604

Offset: 0

N. J. A. Sloane, Nov 13 2009

nonn

			x^7+3*x^31+9*x^55+22*x^79+51*x^103+106*x^127+215*x^151+...

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

Cf. Product_{n>=1} (1 - x^(5*n))^k/(1 - x^n)^(k + 1): A160461 (k=1), this sequence (k=2), A160463 (k=3), A160506 (k=4), A071734 (k=5), A160460 (k=6), A160521 (k=7), A278555 (k=12), A278556 (k=18), A278557 (k=24), A278558 (k=30).

nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))^2/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2016 *)

See Maple code in A160458 for formula.

a(n) ~ sqrt(13/15) * exp(Pi*sqrt(26*n/15)) / (20*n). - Vaclav Kotesovec, Nov 28 2016

A160463 Coefficients in the expansion of C^3/B^4, in Watson's notation of page 106.

Original entry on oeis.org

1, 4, 14, 40, 105, 249, 562, 1198, 2460, 4865, 9352, 17486, 31973, 57220, 100550, 173665, 295413, 495339, 819900, 1340655, 2167825, 3468579, 5495908, 8628080, 13428945, 20730689, 31757174, 48293585, 72933885, 109421095, 163135433, 241763735, 356246552

Offset: 0

N. J. A. Sloane, Nov 13 2009

nonn

			x^11+4*x^35+14*x^59+40*x^83+105*x^107+249*x^131+562*x^155+...

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

Cf. Product_{n>=1} (1 - x^(5*n))^k/(1 - x^n)^(k + 1): A160461 (k=1), A160462 (k=2), this sequence (k=3), A160506 (k=4), A071734 (k=5), A160460 (k=6), A160521 (k=7), A278555 (k=12), A278556 (k=18), A278557 (k=24), A278558 (k=30).

nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))^3/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2016 *)

See Maple code in A160458 for formula.

a(n) ~ sqrt(17/3) * exp(Pi*sqrt(34*n/15)) / (100*n). - Vaclav Kotesovec, Nov 28 2016

A160506 Coefficients in the expansion of C^4/B^5, in Watson's notation of page 106.

Original entry on oeis.org

1, 5, 20, 65, 190, 502, 1245, 2910, 6505, 13965, 29005, 58455, 114810, 220240, 413775, 762635, 1381550, 2463060, 4327445, 7500260, 12836645, 21712470, 36323930, 60143320, 98620425, 160238035, 258110955, 412367705, 653709340, 1028658150, 1607306688

Offset: 0

N. J. A. Sloane, Nov 13 2009

nonn

			x^15+5*x^39+20*x^63+65*x^87+190*x^111+502*x^135+1245*x^159+...

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

Cf. Product_{n>=1} (1 - x^(5*n))^k/(1 - x^n)^(k + 1): A160461 (k=1), A160462 (k=2), A160463 (k=3), this sequence (k=4), A071734 (k=5), A160460 (k=6), A160521 (k=7), A278555 (k=12), A278556 (k=18), A278557 (k=24), A278558 (k=30).

nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))^4/(1 - x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2016 *)

See Maple code in A160458 for formula.

a(n) ~ sqrt(7/5) * exp(Pi*sqrt(14*n/5)) / (100*n). - Vaclav Kotesovec, Nov 28 2016

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A160507 Duplicate of A071734.

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A160461 Coefficients in the expansion of C/B^2, in Watson's notation of page 106.

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

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A160460 Coefficients in the expansion of C^6 / B^7, in Watson's notation of page 106.

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A278556 Expansion of Product_{n>=1} (1 - x^(5*n))^18/(1 - x^n)^19 in powers of x.

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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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A160462 Coefficients in the expansion of C^2/B^3, in Watson's notation of page 106.

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