A280457 -id:A280457 - cp's OEIS frontend

A284499 Expansion of Product_{k>=0} (1 - x^(7*k+1)) in powers of x.

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

Offset: 0

Seiichi Manyama, Mar 28 2017

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. Product_{k>=0} (1 - x^(7*k+m)): this sequence (m=1), A284500 (m=2), A284501 (m=3), A284502 (m=4), A284503 (m=5), A284504 (m=6).

Cf. A280457.

G:= mul(1-x^(7*k+1),k=0..100/7):
S:= series(G,x,101):
seq(coeff(S,x,j),j=0..100); # Robert Israel, Mar 29 2017

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

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

a(n) = -(1/n)*Sum_{k=1..n} A284099(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

A284095 Expansion of Product_{k>=0} (1 + x^(8*k+1)).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Mar 20 2017

nonn

Number of partitions of n into distinct parts congruent to 1 mod 8.

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

Cf. Product_{k>=0} (1 + x^(m*k+1)): A261612 (m=3), A169975 (m=4), A280454 (m=5), A280456 (m=6), A280457 (m=7), this sequence (m=8).

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

Vec(prod(k=0, 91, (1 + x^(8*k + 1))) + O(x^92)) \\ Indranil Ghosh, Mar 20 2017

a(n) ~ exp(sqrt(n/6)*Pi/2) / (2^(15/8) * 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

A284499 Expansion of Product_{k>=0} (1 - x^(7*k+1)) in powers of x.

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

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A284095 Expansion of Product_{k>=0} (1 + x^(8*k+1)).

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cp's OEIS Frontend

A284499 Expansion of Product_{k>=0} (1 - x^(7*k+1)) 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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A284095 Expansion of Product_{k>=0} (1 + x^(8*k+1)).

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