A262928 -id:A262928 - cp's OEIS frontend

A262924 Expansion of Product_{k>=1} (1 + x^(3k-1))^(3k-1) * (1 + x^(3k-2))^(3k-2).

1, 1, 2, 2, 5, 10, 13, 25, 35, 57, 87, 134, 211, 306, 458, 684, 996, 1465, 2129, 3073, 4411, 6288, 8977, 12707, 17913, 25185, 35231, 49078, 68228, 94490, 130408, 179425, 246121, 336681, 459239, 624842, 847986, 1147728, 1549773, 2087972, 2806455, 3764136

Offset: 0

Vaclav Kotesovec, Oct 04 2015

nonn

Convolution of A262948 and A262949.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vaclav Kotesovec)
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Cf. A262884, A262878, A262879, A262923, A261612, A262928, A262948, A262949.

Cf. A262736, A285292, A285293.

nmax=60; CoefficientList[Series[Product[(1 + x^(3*k-1))^(3*k-1)*(1 + x^(3*k-2))^(3*k-2), {k, 1, nmax}], {x, 0, nmax}], x]

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

A280456 Expansion of Product_{k>=0} (1 + x^(6*k+1)).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 3, 2, 0, 0, 0, 1, 3, 3, 1, 0, 0, 1, 4, 4, 1, 0, 0, 1, 4, 5, 2, 0, 0, 1, 5, 7, 3, 0, 0, 1, 5, 8, 5, 1, 0, 1, 6, 10, 6, 1, 0, 1, 6, 12, 9, 2, 0, 1, 7, 14, 11, 3, 0, 1, 7, 16, 15, 5, 0, 1, 8, 19, 18, 7, 1, 1, 8, 21, 23, 10, 1, 1, 9, 24

Offset: 0

Ilya Gutkovskiy, Jan 03 2017

nonn

Number of partitions of n into distinct parts congruent to 1 mod 6.

Convolution of A281244 and A280456 is A098884. - Vaclav Kotesovec, Jan 18 2017

			a(32) = 3 because we have [31, 1], [25, 7] and [19, 13].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015.
Index entries for related partition-counting sequences

Cf. A000700, A016921, A109701, A169975, A261612, A280454, A280457.

Cf. A262928, A147599, A281243, A281244, A281245.

nmax = 105; CoefficientList[Series[Product[(1 + x^(6 k + 1)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[If[Mod[k, 6] == 1, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 18 2017 *)

G.f.: Product_{k>=0} (1 + x^(6*k+1)).

a(n) ~ exp(Pi*sqrt(n)/(3*sqrt(2)))/(2*2^(5/12)*sqrt(3)*n^(3/4)) * (1 + (Pi/(144*sqrt(2)) - 9/(4*sqrt(2)*Pi)) / sqrt(n)). - Ilya Gutkovskiy, Jan 03 2017, extended by Vaclav Kotesovec, Jan 18 2017

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)).

A262948 Expansion of Product_{k>=1} (1 + x^(3k-1))^(3k-1).

Original entry on oeis.org

1, 0, 2, 0, 1, 5, 0, 10, 8, 5, 26, 11, 28, 62, 24, 101, 111, 77, 260, 202, 268, 583, 382, 761, 1165, 847, 1940, 2198, 2061, 4346, 4084, 5078, 9039, 7844, 11978, 17620, 15721, 26648, 33219, 32894, 56000, 61494, 69653, 111884, 114265, 146557, 214864, 214967

Offset: 0

Vaclav Kotesovec, Oct 05 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A262736, A262878, A262884, A262924, A262928, A262949.

nmax=60; CoefficientList[Series[Product[(1 + x^(3*k-1))^(3*k-1),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ exp(3 * 2^(-4/3) * Zeta(3)^(1/3) * n^(2/3)) * Zeta(3)^(1/6) / (2^(7/12) * sqrt(3*Pi) * n^(2/3)).

A284315 Expansion of Product_{k>=0} (1 - x^(3*k+2)) in powers of x.

Original entry on oeis.org

1, 0, -1, 0, 0, -1, 0, 1, -1, 0, 1, -1, 0, 2, -1, -1, 2, -1, -1, 3, -1, -2, 3, -1, -3, 4, 0, -4, 4, 0, -5, 5, 1, -7, 5, 2, -8, 6, 4, -10, 5, 5, -12, 6, 8, -14, 5, 10, -16, 5, 14, -19, 3, 17, -21, 2, 22, -23, -1, 26, -26, -3, 33, -28, -8, 38, -30, -12, 46, -32, -19

Offset: 0

Seiichi Manyama, Mar 25 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. Product_{k>=0} (1 - x^(m*k+m-1)): A081362 (m=2), this sequence (m=3), A284316 (m=4), A284317 (m=5).

Cf. A078182, A262928.

CoefficientList[Series[Product[1 - x^(3k + 2), {k, 0, 100}], {x, 0, 100}], x] (* Indranil Ghosh, Mar 25 2017 *)

Vec(prod(k=0, 100, 1 - x^(3*k+2)) + O(x^101)) \\ Indranil Ghosh, Mar 25 2017

a(n) = -(1/n) * Sum_{k=1..n} A078182(k) * a(n-k), a(0) = 1.

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).

A374065 Expansion of Product_{k>=1} 1 / (1 + x^(3*k-2)).

Original entry on oeis.org

1, -1, 1, -1, 0, 0, 0, -1, 2, -2, 1, 0, -1, 0, 2, -3, 3, -1, -1, 1, 1, -4, 5, -3, 0, 2, 0, -4, 7, -6, 1, 3, -2, -3, 9, -10, 4, 3, -5, -1, 11, -15, 10, 1, -8, 3, 10, -20, 17, -3, -10, 9, 7, -24, 26, -10, -10, 15, 2, -27, 37, -21, -8, 22, -6, -28, 49, -36, -2, 30, -19, -24, 61, -56, 10, 35

Offset: 0

Ilya Gutkovskiy, Jun 27 2024

sign

Cf. A035382, A081360, A081362, A109389, A132462, A261612, A262928, A284312, A374064.

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

a(0) = 1; a(n) = -Sum_{k=1..n} A261612(k) * a(n-k).
a(n) = Sum_{k=0..n} A081362(k) * A132462(n-k).
a(n) = Sum_{k=0..n} A109389(k) * A262928(n-k).

A262952 Expansion of Product_{k>=1} (1 + x^(2k-1)) (1 + x^(3*k-1)).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 3, 3, 6, 5, 7, 9, 9, 12, 15, 16, 21, 24, 26, 33, 37, 42, 51, 57, 65, 78, 86, 99, 115, 128, 146, 168, 187, 213, 243, 269, 306, 345, 383, 433, 487, 539, 607, 678, 749, 842, 935, 1033, 1157, 1279, 1413, 1575, 1736, 1916, 2127, 2339, 2579, 2853

Offset: 0

Vaclav Kotesovec, Oct 05 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Cf. A000700, A262928, A262953.

nmax = 60; CoefficientList[Series[Product[(1 + x^(2*k-1)) * (1 + x^(3*k-1)), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ 5^(1/4) * exp(Pi*sqrt(5*n/2)/3) / (2^(23/12) * sqrt(3) * n^(3/4)).

A284093 Expansion of Product_{k>=1} (1 + x^(8*k-1)).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 0, 2, 3, 1, 0, 0, 0, 0, 0, 3, 4, 1, 0, 0, 0, 0, 1, 4, 4, 1, 0, 0, 0, 0, 1, 5, 5, 1

Offset: 0

Seiichi Manyama, Mar 20 2017

nonn

Number of partitions into distinct parts 8*k-1.

Vaclav Kotesovec, Table of n, a(n) for n = 0..20000

Cf. Product_{k>=1} (1 + x^(m*k-1)): A262928 (m=3), A147599 (m=4), A281243 (m=5), A281244 (m=6), A281245 (m=7), this sequence (m=8).

CoefficientList[Series[Product[(1 + x^(8*k - 1)) , {k, 1, 91}], {x, 0, 91}], x] (* Indranil Ghosh, Mar 20 2017 *)
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 8] == 7, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Mar 20 2017 *)

Vec(prod(k=1, 91, (1 + x^(8*k - 1))) + O(x^92)) \\ Indranil Ghosh, Mar 20 2017

a(n) ~ exp(sqrt(n/6)*Pi/2) / (2^(21/8) * 3^(1/4) * n^(3/4)) * (1 + (11*Pi/(384*sqrt(6)) - 3*sqrt(3/2)/(2*Pi))/sqrt(n)). - Vaclav Kotesovec, Mar 20 2017

G.f.: Sum_{k>=0} x^(k*(4*k + 3)) / Product_{j=1..k} (1 - x^(8*j)). - Ilya Gutkovskiy, Nov 24 2020

cp's OEIS Frontend

A262924 Expansion of Product_{k>=1} (1 + x^(3*k-1))^(3*k-1) * (1 + x^(3*k-2))^(3*k-2).

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A280456 Expansion of Product_{k>=0} (1 + x^(6*k+1)).

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

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A262948 Expansion of Product_{k>=1} (1 + x^(3*k-1))^(3*k-1).

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A284315 Expansion of Product_{k>=0} (1 - x^(3*k+2)) in powers of x.

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

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A374065 Expansion of Product_{k>=1} 1 / (1 + x^(3*k-2)).

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A262952 Expansion of Product_{k>=1} (1 + x^(2*k-1)) * (1 + x^(3*k-1)).

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A284093 Expansion of Product_{k>=1} (1 + x^(8*k-1)).

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A262924 Expansion of Product_{k>=1} (1 + x^(3k-1))^(3k-1) * (1 + x^(3k-2))^(3k-2).

A262948 Expansion of Product_{k>=1} (1 + x^(3k-1))^(3k-1).

A262952 Expansion of Product_{k>=1} (1 + x^(2k-1)) (1 + x^(3*k-1)).