A035386 -id:A035386 - cp's OEIS frontend

A109708 Number of partitions of n into parts each equal to 6 mod 7.

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 2, 1, 0, 0, 1, 1, 2, 2, 1, 0, 1, 1, 2, 3, 3, 1, 1, 1, 2, 3, 4, 3, 2, 1, 2, 3, 5, 5, 5, 2, 2, 3, 5, 6, 8, 5, 3, 3, 5, 7, 10, 9, 7, 4, 5, 7, 11, 12, 12, 8, 6, 7, 12, 14, 17, 15, 11, 8, 12, 15, 20, 21, 19, 13, 13, 16, 22, 26, 28, 23

Offset: 0

Erich Friedman, Aug 07 2005

nonn

			a(45)=3 because we have 45=27+6+6+6=20+13+6+6=13+13+13+6.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A284105.

Cf. similar sequences of number of partitions of n into parts congruent to m-1 mod m: A000009 (m=2), A035386 (m=3), A035462 (m=4), A109700 (m=5), A109702 (m=6), this sequence (m=7).

g:=1/product(1-x^(6+7*j),j=0..20): gser:=series(g,x=0,98): seq(coeff(gser,x,n),n=0..95); # Emeric Deutsch, Apr 14 2006

nmax=100; CoefficientList[Series[Product[1/(1-x^(7*k+6)),{k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 27 2015 *)

G.f.: 1/product(1-x^(6+7j), j=0..infinity). - Emeric Deutsch, Apr 14 2006
a(n) ~ Gamma(6/7) * exp(Pi*sqrt(2*n/21)) / (2^(27/14) * 3^(3/7) * 7^(1/14) * Pi^(1/7) * n^(13/14)) * (1 - (39*sqrt(3/14)/(7*Pi) + 13*Pi/(168*sqrt(42))) / sqrt(n)). - Vaclav Kotesovec, Feb 27 2015, extended Jan 24 2017
a(n) = (1/n)*Sum_{k=1..n} A284105(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 20 2017

Changed offset to 0 and added a(0)=1 by Vaclav Kotesovec, Feb 27 2015

A337548 Number of compositions (ordered partitions) of n into distinct parts congruent to 2 mod 3.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 2, 1, 0, 2, 1, 0, 4, 1, 6, 4, 1, 6, 6, 1, 12, 6, 1, 18, 8, 25, 24, 8, 25, 30, 10, 49, 42, 10, 73, 48, 12, 121, 60, 132, 145, 72, 134, 217, 84, 254, 265, 96, 376, 361, 114, 616, 433, 126, 858, 553, 864, 1218, 649, 882, 1580, 817, 1620, 2180, 937

Offset: 0

Ilya Gutkovskiy, Nov 22 2020

nonn

			a(15) = 6 because we have [8, 5, 2], [8, 2, 5], [5, 8, 2], [5, 2, 8], [2, 8, 5] and [2, 5, 8].

Index entries for sequences related to compositions

Cf. A000931, A016789, A032020, A032021, A035386, A262928, A337547, A339059, A339060.

nmax = 65; CoefficientList[Series[Sum[k! x^(k (3 k + 1)/2)/Product[1 - x^(3 j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

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

A374064 Expansion of Product_{k>=1} 1 / (1 + x^(3*k-1)).

Original entry on oeis.org

1, 0, -1, 0, 1, -1, -1, 1, 0, -1, 1, 0, -1, 1, 0, -2, 2, 1, -3, 1, 3, -3, 0, 3, -3, -1, 4, -3, -1, 5, -3, -3, 7, -3, -5, 7, -1, -7, 8, 0, -8, 8, 1, -11, 10, 3, -14, 9, 8, -17, 8, 10, -18, 6, 14, -22, 6, 19, -24, 1, 26, -26, -3, 30, -25, -9, 37, -27, -13, 42, -26, -23, 51, -25, -31, 56

Offset: 0

Ilya Gutkovskiy, Jun 27 2024

sign

Cf. A035386, A081360, A081362, A109389, A132463, A261612, A262928, A284315, A374065.

nmax = 75; CoefficientList[Series[Product[1/(1 + x^(3 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[DivisorSum[k, (-1)^(k/#) # &, Mod[#, 3] == 2 &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

a(0) = 1; a(n) = -Sum_{k=1..n} A262928(k) * a(n-k).
a(n) = Sum_{k=0..n} A081362(k) * A132463(n-k).
a(n) = Sum_{k=0..n} A109389(k) * A261612(n-k).

A374058 Expansion of Product_{k>=1} (1 - x^(3k-2)) (1 - x^(3*k)).

Original entry on oeis.org

1, -1, 0, -1, 0, 1, -1, 1, 0, 0, 1, 0, -1, 1, -1, 1, 0, -1, 0, 0, -1, 1, 0, -1, 1, -1, 0, 1, -1, 0, 1, -1, 1, 1, -1, 0, 0, -1, 2, 0, -1, 1, 0, -1, 2, -2, 0, 1, -1, 0, 1, -1, 0, 1, -2, 1, 1, -2, 1, 0, -2, 2, 0, -2, 2, -1, 0, 2, -1, -1, 1, -1, -1, 3, -2, 0, 2, -2, 1, 2, -3, 1, 1, -2, 2, 1

Offset: 0

Ilya Gutkovskiy, Jun 27 2024

sign

Ilya Gutkovskiy, Scatterplot of a(n) up to n=10000

Cf. A010815, A035360, A035386, A081362, A082051, A132463, A137569, A274719, A284312, A374060.

nmax = 85; CoefficientList[Series[Product[(1 - x^(3 k - 2)) (1 - x^(3 k)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -(1/n) Sum[Plus @@ Select[Divisors[k], Mod[#, 3] != 2 &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 85}]

a(0) = 1; a(n) = -(1/n) * Sum_{k=1..n} A082051(k) * a(n-k).
a(0) = 1; a(n) = -Sum_{k=1..n} A035360(k) * a(n-k).
a(n) = Sum_{k=0..n} A010815(k) * A035386(n-k).

A362697 Expansion of e.g.f. Product_{k>0} (1 - x^(3k-1))^(-1/(3k-1)).

Original entry on oeis.org

1, 0, 1, 0, 9, 24, 225, 504, 16065, 27216, 1555281, 6123600, 159249321, 779262120, 31816914129, 240363179784, 8207359913025, 66059979227424, 2145292484152545, 19782668403572256, 1015331126023222281, 7961977144683689400, 454920488042137314561

Offset: 0

Seiichi Manyama, Jul 07 2023

nonn

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

Cf. A001822, A035386, A206303, A262946, A362696.

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

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} A001822(k) * a(n-k)/(n-k)!.

A304883 Expansion of Product_{k>=1} 1/(1-x^(3k-1)) Product_{k>=1} 1/(1-x^(6*k-5)).

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 7, 9, 11, 14, 17, 21, 26, 32, 39, 47, 56, 67, 80, 95, 113, 133, 156, 183, 214, 250, 291, 338, 391, 452, 521, 600, 690, 791, 906, 1035, 1181, 1346, 1532, 1741, 1975, 2238, 2532, 2862, 3231, 3643, 4103, 4615, 5186, 5822, 6529, 7315, 8187, 9154

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. A035386, A109701, A304885.

seq(coeff(series(mul(1/(1-x^(3*k-1)),k=1..n)*mul(1/(1-x^(6*k-5)),k=1..n), x,70),x,n),n=0..60); # Muniru A Asiru, May 21 2018

CoefficientList[ Series[ Product[1/(1 - x^(3k -1)), {k, 18}]*Product[1/(1 - x^(6k -5)), {k, 9}], {x, 0, 54}], x] (* Robert G. Wilson v, May 20 2018 *)

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

a(n) ~ exp(Pi*sqrt(n/3)) * Gamma(1/3) / (4 * 3^(1/3) * Pi^(2/3) * n^(2/3)). - Vaclav Kotesovec, May 21 2018

A374018 Expansion of Product_{k>=1} 1 / (1 - x^(3*k-1))^2.

Original entry on oeis.org

1, 0, 2, 0, 3, 2, 4, 4, 7, 6, 13, 10, 19, 18, 27, 30, 42, 44, 63, 66, 91, 100, 130, 144, 187, 206, 263, 294, 364, 412, 506, 568, 696, 782, 943, 1070, 1273, 1444, 1713, 1936, 2285, 2586, 3027, 3428, 3996, 4516, 5243, 5924, 6841, 7730, 8895, 10030, 11512, 12966, 14825, 16696

Offset: 0

Ilya Gutkovskiy, Jun 25 2024

nonn

Cf. A000712, A022567, A035386, A078182, A261616, A374019.

nmax = 55; CoefficientList[Series[Product[1/(1 - x^(3 k - 1))^2, {k, 1, nmax}], {x, 0, nmax}], x]

a(0) = 1; a(n) = (2/n) * Sum_{k=1..n} A078182(k) * a(n-k).
a(n) = Sum_{k=0..n} A035386(k) * A035386(n-k).
a(n) ~ exp(2*Pi*sqrt(n)/3) * Pi^(4/3) / (3^(3/2) * Gamma(1/3)^2 * n^(11/12)). - Vaclav Kotesovec, Jun 25 2024

cp's OEIS Frontend

A109708 Number of partitions of n into parts each equal to 6 mod 7.

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A337548 Number of compositions (ordered partitions) of n into distinct parts congruent to 2 mod 3.

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A374064 Expansion of Product_{k>=1} 1 / (1 + x^(3*k-1)).

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A374058 Expansion of Product_{k>=1} (1 - x^(3*k-2)) * (1 - x^(3*k)).

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A362697 Expansion of e.g.f. Product_{k>0} (1 - x^(3*k-1))^(-1/(3*k-1)).

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A304883 Expansion of Product_{k>=1} 1/(1-x^(3*k-1)) * Product_{k>=1} 1/(1-x^(6*k-5)).

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Maple

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A374018 Expansion of Product_{k>=1} 1 / (1 - x^(3*k-1))^2.

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Mathematica

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A374058 Expansion of Product_{k>=1} (1 - x^(3k-2)) (1 - x^(3*k)).

A362697 Expansion of e.g.f. Product_{k>0} (1 - x^(3k-1))^(-1/(3k-1)).

A304883 Expansion of Product_{k>=1} 1/(1-x^(3k-1)) Product_{k>=1} 1/(1-x^(6*k-5)).