A281245 -id:A281245 - cp's OEIS frontend

A281458 Expansion of Product_{k>=1} (1 + x^(7*k-5)).

1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 3, 0, 2, 0, 0, 1, 0, 3, 0, 3, 0, 1, 1, 0, 4, 0, 4, 0, 1, 1, 0, 4, 0, 5, 0, 2, 1, 0, 5, 0, 7, 0, 3, 1, 0, 5, 0, 8, 0, 5, 1, 1, 6, 0, 10, 0, 6

Offset: 0

Vaclav Kotesovec, Jan 22 2017

nonn

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

Cf. A109704, A281245, A281455, A281456, A281457, A280457.

nmax = 100; CoefficientList[Series[Product[(1 + x^(7*k - 5)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 7] == 2, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly

a(n) ~ exp(sqrt(n/21)*Pi) / (2^(9/7)*21^(1/4)*n^(3/4)) * (1 - (3*sqrt(21)/(8*Pi) + 11*Pi/(336*sqrt(21))) / sqrt(n)). - Vaclav Kotesovec, Jan 22 2017, extended Jan 24 2017

A284504 Expansion of Product_{k>=0} (1 - x^(7*k+6)) in powers of x.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 2, -1, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0, -1, 3, -1, 0, 0, 0, 0, -2, 3, -1, 0, 0, 0, 0, -3, 4, -1, 0, 0, 0, 1, -4, 4, -1, 0, 0, 0, 1, -5, 5, -1, 0, 0, 0, 2, -7, 5

Offset: 0

Seiichi Manyama, Mar 28 2017

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

Cf. Product_{k>=0} (1 - x^(7*k+m)): A284499 (m=1), A284500 (m=2), A284501 (m=3), A284502 (m=4), A284503 (m=5), this sequence (m=6).

Cf. A281245.

CoefficientList[Series[Product[1 - x^(7k + 6), {k, 0, 100}], {x, 0, 100}], x] (* Indranil Ghosh, Mar 28 2017 *)

Vec(prod(k=0, 100, 1 - x^(7*k + 6)) + O(x^101)) \\ Indranil Ghosh, Mar 28 2017

a(n) = -(1/n) * Sum_{k=1..n} A284105(k) * a(n-k), a(0) = 1.

A281459 Expansion of Product_{k>=1} (1 + x^(7k-1))(1 + x^(7*k-6)).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 3, 3, 2, 1, 0, 1, 3, 5, 5, 3, 1, 0, 2, 5, 7, 7, 5, 2, 1, 3, 7, 11, 11, 7, 3, 2, 5, 11, 15, 15, 11, 5, 3, 7, 15, 22, 22, 15, 7, 5, 11, 22, 30, 30, 22, 12, 8, 15, 30, 42, 42, 30, 16, 12, 23, 42, 56

Offset: 0

Vaclav Kotesovec, Jan 22 2017

nonn

Convolution of A281245 and A280457.

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

Cf. A035430, A281245, A280457.

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

a(n) ~ exp(sqrt(2*n/21)*Pi) / (2^(5/4)*21^(1/4)*n^(3/4)) * (1 + (13*Pi/(84*sqrt(42)) - 3*sqrt(21/2)/(8*Pi)) / sqrt(n)). - Vaclav Kotesovec, Jan 22 2017, extended Jan 24 2017

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

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A281458 Expansion of Product_{k>=1} (1 + x^(7*k-5)).

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A284504 Expansion of Product_{k>=0} (1 - x^(7*k+6)) in powers of x.

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A281459 Expansion of Product_{k>=1} (1 + x^(7*k-1))*(1 + x^(7*k-6)).

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

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A281459 Expansion of Product_{k>=1} (1 + x^(7k-1))(1 + x^(7*k-6)).