A035382 -id:A035382 - cp's OEIS frontend

A277210 Expansion of Product_{k>=1} 1/(1 - x^(3*k+1)).

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 3, 3, 4, 5, 4, 6, 6, 7, 7, 9, 8, 11, 11, 12, 13, 16, 15, 18, 20, 22, 22, 27, 27, 31, 33, 37, 38, 45, 46, 51, 55, 62, 63, 72, 76, 84, 89, 99, 103, 116, 122, 133, 142, 158, 164, 181, 193, 210, 222, 245, 257, 281, 299, 324, 343, 376, 396, 429, 457, 495, 522, 568, 601, 649, 689

Offset: 0

Ilya Gutkovskiy, Oct 05 2016

nonn

Number of partitions of n into parts larger than 1 and congruent to 1 mod 3.

More generally, the ordinary generating function for the number of partitions of n into parts larger than 1 and congruent to 1 mod m (for m>0) is Product_{k>=1} 1/(1 - x^(m*k+1)).

			a(14) = 2, because we have [10, 4] and [7, 7].

Index entries for related partition-counting sequences

Cf. A016777, A035382, A087897, A117957.

CoefficientList[Series[(1 - x)/QPochhammer[x, x^3], {x, 0, 85}], x]

G.f.: Product_{k>=1} 1/(1 - x^(3*k+1)).

a(n) ~ Pi^(1/3) * Gamma(1/3) * exp(sqrt(2*n)*Pi/3) / (2^(13/6)*3^(3/2)*n^(7/6)). - Vaclav Kotesovec, Oct 06 2016

A362696 Expansion of e.g.f. Product_{k>0} (1 - x^(3k-2))^(-1/(3k-2)).

Original entry on oeis.org

1, 1, 2, 6, 30, 150, 900, 7020, 62460, 562140, 5984280, 67252680, 863165160, 11700148680, 173098134000, 2625661170000, 45310413258000, 782198417206800, 14310269286746400, 280333959468789600, 6002139207488767200, 129820528515538159200, 2934651197018947982400

Offset: 0

Seiichi Manyama, Jul 07 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A001817, A035382, A206303, A262947, A362697.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1,N, (1-x^(3*k-2))^(1/(3*k-2)))))

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} A001817(k) * a(n-k)/(n-k)!.

A261631 Expansion of Product_{k>=0} 1/(1-x^(3*k+1))^3.

Original entry on oeis.org

1, 3, 6, 10, 18, 30, 46, 69, 105, 154, 219, 309, 434, 597, 813, 1100, 1476, 1959, 2585, 3387, 4410, 5709, 7353, 9414, 12001, 15231, 19242, 24205, 30348, 37902, 47165, 58500, 72342, 89169, 109599, 134337, 164221, 200226, 243537, 295496, 357732, 432117, 520858

Offset: 0

Vaclav Kotesovec, Aug 27 2015

nonn

Cf. A035382, A261616, A261632.

nmax=50; CoefficientList[Series[Product[1/(1-x^(3*k+1))^3, {k, 0, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt(2*n/3)) * Gamma(1/3)^3 / (4 * Pi^2 * sqrt(6*n)).

A261635 Expansion of Product_{k>=0} 1/(1-x^(3*k+1))^4.

Original entry on oeis.org

1, 4, 10, 20, 39, 72, 124, 204, 331, 524, 806, 1216, 1813, 2660, 3846, 5500, 7790, 10916, 15158, 20880, 28544, 38736, 52226, 69972, 93200, 123460, 162700, 213340, 278459, 361860, 468252, 603484, 774844, 991220, 1263576, 1605392, 2033172, 2566972, 3231338

Offset: 0

Vaclav Kotesovec, Aug 27 2015

nonn

Cf. A035382, A261616, A261636.

nmax=50; CoefficientList[Series[Product[1/(1-x^(3*k+1))^4, {k, 0, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(2*Pi*sqrt(2*n)/3) * Gamma(1/3)^4 / (8 * 2^(1/12) * sqrt(3) * Pi^(8/3) * n^(5/12)).

A304885 Expansion of Product_{k>=1} 1/(1-x^(3k-2)) Product_{k>=1} 1/(1-x^(6*k-1)).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 3, 4, 5, 6, 8, 10, 12, 14, 17, 21, 25, 30, 35, 41, 49, 58, 68, 79, 92, 107, 124, 144, 166, 191, 220, 252, 289, 331, 378, 431, 490, 557, 632, 717, 812, 917, 1035, 1167, 1315, 1480, 1663, 1866, 2092, 2344, 2624, 2934, 3277, 3656, 4076, 4542, 5056

Offset: 0

Seiichi Manyama, May 20 2018

nonn

Sylvie Corteel, Carla D. Savage, and Andrew V. Sills. F. Beukers, Lecture hall sequences, q-series, and asymmetric partition identities, In Alladi K., Garvan F. (eds) Partitions, q-Series, and Modular Forms pp 53-68. Developments in Mathematics, vol 23. Springer, New York, NY.

Cf. A035382, A109702, A304883.

seq(coeff(series(mul(1/(1-x^(3*k-2)),k=1..n)*mul(1/(1-x^(6*k-1)),k=1..n), x,70),x,n),n=0..60); # Muniru A Asiru, May 21 2018

nmax = 50; CoefficientList[Series[Product[1/((1-x^(3*k-2)) * (1-x^(6*k-1))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 21 2018 *)

G.f.: Sum_{j>=0} x^(j*(3*j-1)/2)*(Product_{k=1..j} (1-x^(6*k-4)))/(Product_{k=1..3*j} (1-x^k)).

a(n) ~ exp(Pi*sqrt(n/3)) * Pi^(2/3) / (2 * 3^(2/3) * Gamma(1/3) * n^(5/6)). - Vaclav Kotesovec, May 21 2018

cp's OEIS Frontend

A277210 Expansion of Product_{k>=1} 1/(1 - x^(3*k+1)).

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A362696 Expansion of e.g.f. Product_{k>0} (1 - x^(3*k-2))^(-1/(3*k-2)).

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A261631 Expansion of Product_{k>=0} 1/(1-x^(3*k+1))^3.

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A261635 Expansion of Product_{k>=0} 1/(1-x^(3*k+1))^4.

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A304885 Expansion of Product_{k>=1} 1/(1-x^(3*k-2)) * Product_{k>=1} 1/(1-x^(6*k-1)).

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A362696 Expansion of e.g.f. Product_{k>0} (1 - x^(3k-2))^(-1/(3k-2)).

A304885 Expansion of Product_{k>=1} 1/(1-x^(3k-2)) Product_{k>=1} 1/(1-x^(6*k-1)).