A147599 -id:A147599 - cp's OEIS frontend

A284316 Expansion of Product_{k>=0} (1 - x^(4*k+3)) in powers of x.

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

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), A284315 (m=3), this sequence (m=4), A284317 (m=5).

Cf. A050452, A147599, A284313, A111330.

CoefficientList[Series[Product[1 - x^(4k + 3), {k, 0, 100}], {x, 0, 100}], x] (* Indranil Ghosh, Mar 25 2017 *)

Vec(prod(k=0, 100, 1 - x^(4*k+3)) + O(x^101)) \\ Indranil Ghosh, Mar 25 2017

a(n) = -(1/n)*Sum_{k=1..n} A050452(k)*a(n-k), a(0) = 1.

O.g.f.: Sum_{n >= 0} (-1)^n*x^(n*(2*n+1)) / Product_{k = 1..n} ( 1 - x^(4*k) ). Cf. A284313. - Peter Bala, Nov 28 2020

A301505 Expansion of Product_{k>=1} (1 + x^(4k))(1 + x^(4*k-1)).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 1, 3, 2, 0, 2, 5, 2, 0, 4, 7, 3, 1, 7, 10, 4, 2, 11, 14, 5, 4, 17, 19, 6, 8, 25, 25, 9, 13, 36, 33, 12, 21, 50, 43, 16, 33, 69, 55, 23, 49, 93, 70, 32, 71, 124, 89, 45, 102, 163, 112, 64, 142, 212, 141, 89, 195, 273, 177, 123, 265, 349

Offset: 0

Ilya Gutkovskiy, Mar 22 2018

nonn

Number of partitions of n into distinct parts congruent to 0 or 3 mod 4.

			a(11) = 3 because we have [11], [8, 3] and [7, 4].

Index entries for sequences related to partitions

Cf. A014601, A035364, A035457, A131795, A147599, A301504, A301507, A301508.

nmax = 70; CoefficientList[Series[Product[(1 + x^(4 k)) (1 + x^(4 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[x QPochhammer[-1, x^4] QPochhammer[-x^(-1), x^4]/(2 (1 + x)), {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[Product[(1 + Boole[MemberQ[{0, 3}, Mod[k, 4]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A014601(k)).

a(n) ~ exp(Pi*sqrt(n/6)) / (2^(5/2)*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Mar 23 2018

A301508 Expansion of Product_{k>=0} (1 + x^(4k+2))(1 + x^(4*k+3)).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 2, 3, 4, 4, 5, 5, 6, 7, 6, 8, 9, 9, 11, 12, 13, 14, 15, 17, 19, 20, 23, 25, 27, 29, 31, 35, 37, 40, 46, 48, 52, 57, 60, 66, 71, 76, 85, 90, 97, 105, 112, 121, 129, 140, 152, 161, 174, 187, 198, 214, 228, 245, 265, 280, 302, 323, 342

Offset: 0

Ilya Gutkovskiy, Mar 22 2018

nonn

Number of partitions of n into distinct parts congruent to 2 or 3 mod 4.

			a(13) = 3 because we have [11, 2], [10, 3] and [7, 6].

Index entries for sequences related to partitions

Cf. A035366, A042964, A070048, A131795, A147599, A301504, A301505, A301507.

nmax = 70; CoefficientList[Series[Product[(1 + x^(4 k + 2)) (1 + x^(4 k + 3)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[QPochhammer[-x^2, x^4] QPochhammer[-x^3, x^4], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[Product[(1 + Boole[MemberQ[{2, 3}, Mod[k, 4]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A042964(k)).

a(n) ~ exp(Pi*sqrt(n/6)) / (4*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Mar 23 2018

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

A170960 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 7.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Aug 29 2010

Product_{i=1..m} (1 + x^(4*i-1)) is the Poincaré polynomial for both Sp(2m) and O(2m+1).

H. Weyl, The Classical Groups, Princeton, 1946, see p. 238.

Nathaniel Johnston, Table of n, a(n) for n = 0..105 (full sequence)

Cf. A000700, A147599, A169975, A170956-A170965.

a(n) = a(105-n). - Rick L. Shepherd, Mar 01 2013

A170961 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 8.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Aug 29 2010

Product_{i=1..m} (1 + x^(4*i-1)) is the Poincaré polynomial for both Sp(2m) and O(2m+1).

H. Weyl, The Classical Groups, Princeton, 1946, see p. 238.

Nathaniel Johnston, Table of n, a(n) for n = 0..136 (full sequence)

Cf. A000700, A147599, A169975, A170956-A170965.

CoefficientList[Series[Product[1+x^(4i-1),{i,8}],{x,0,110}],x] (* Harvey P. Dale, Aug 22 2012 *)

a(n) = a(136-n). - Rick L. Shepherd, Mar 01 2013

A170962 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 9.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 1, 2, 1, 0, 1, 3, 1, 0, 2, 3, 1, 0, 3, 4, 1, 1, 4, 4, 0, 1, 5, 4, 0, 2, 7, 3, 0, 3, 7, 3, 0, 5, 8, 2, 1, 6, 8, 2, 1, 8, 8, 1, 2, 9, 7, 1, 3, 11, 7, 0, 5, 11, 5, 0, 6, 12, 4, 1, 8, 11, 3, 1, 9, 11, 2, 2, 11, 9, 1, 3, 11, 8, 1, 4, 12, 6, 0, 5, 11, 5

Offset: 0

N. J. A. Sloane, Aug 29 2010

Product_{i=1..m} (1 + x^(4*i-1)) is the Poincaré polynomial for both Sp(2m) and O(2m+1).

H. Weyl, The Classical Groups, Princeton, 1946, see p. 238.

Nathaniel Johnston, Table of n, a(n) for n = 0..171 (full sequence)

Cf. A000700, A147599, A169975, A170956-A170965.

a(n) = a(171-n). - Rick L. Shepherd, Mar 01 2013

A170963 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 10.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 1, 2, 1, 0, 1, 3, 1, 0, 2, 3, 1, 0, 3, 4, 1, 1, 4, 4, 1, 1, 5, 5, 0, 2, 7, 4, 0, 3, 8, 4, 0, 5, 9, 3, 1, 6, 10, 3, 1, 9, 10, 2, 2, 10, 10, 2, 3, 13, 10, 1, 5, 14, 9, 1, 7, 16, 8, 1, 9, 16, 7, 1, 11, 18, 5, 2, 14, 16, 4, 3, 16, 16, 3, 5, 18, 14

Offset: 0

N. J. A. Sloane, Aug 29 2010

Product_{i=1..m} (1 + x^(4*i-1)) is the Poincaré polynomial for both Sp(2m) and O(2m+1).

H. Weyl, The Classical Groups, Princeton, 1946, see p. 238.

Nathaniel Johnston, Table of n, a(n) for n = 0..210 (full sequence)

Cf. A000700, A147599, A169975, A170956-A170965.

a(n) = a(210-n). - Rick L. Shepherd, Mar 01 2013

A170964 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 11.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 1, 2, 1, 0, 1, 3, 1, 0, 2, 3, 1, 0, 3, 4, 1, 1, 4, 4, 1, 1, 5, 5, 1, 2, 7, 5, 0, 3, 8, 5, 0, 5, 10, 4, 1, 6, 11, 4, 1, 9, 12, 3, 2, 11, 12, 3, 3, 14, 13, 2, 5, 16, 12, 2, 7, 19, 12, 2, 10, 20, 11, 2, 12, 23, 10, 2, 16, 23, 8, 3, 19, 24, 7, 5

Offset: 0

N. J. A. Sloane, Aug 29 2010

Product_{i=1..m} (1 + x^(4*i-1)) is the Poincaré polynomial for both Sp(2m) and O(2m+1).

H. Weyl, The Classical Groups, Princeton, 1946, see p. 238.

Nathaniel Johnston, Table of n, a(n) for n = 0..253 (full sequence)

Cf. A000700, A147599, A169975, A170956-A170965.

CoefficientList[Series[Product[1+x^(4k-1),{k,11}],{x,0,100}],x] (* Harvey P. Dale, Sep 19 2020 *)

a(n) = a(253-n). - Rick L. Shepherd, Mar 01 2013

A309240 Expansion of 1/((1 - x)(1 - x^2)(1 + x^3)(1 + x^4)(1 - x^5)(1 - x^6)(1 + x^7)(1 + x^8)...).

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 3, 3, 4, 2, 4, 4, 7, 5, 7, 6, 11, 9, 13, 10, 17, 14, 20, 15, 25, 22, 32, 24, 36, 31, 48, 38, 55, 45, 68, 55, 79, 65, 97, 79, 112, 91, 136, 113, 159, 128, 186, 156, 221, 179, 256, 213, 301, 245, 347, 290, 409, 334, 466, 388, 547, 451, 624, 517, 724, 600, 828, 687, 955, 793, 1088

Offset: 0

Ilya Gutkovskiy, Jul 17 2019

nonn

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

Cf. A000700, A035451, A147599, A300574.

nmax = 70; CoefficientList[Series[Product[1/(1 + (-1)^(k (k + 1)/2) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[Product[(1 + x^(4 k - 2))/((1 + x^(4 k - 1)) (1 - x^(4 k - 3))), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[2/(QPochhammer[-1, -x^2] QPochhammer[x, -x^2]), {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 + (-1)^(k*(k+1)/2) * x^k).
G.f.: Product_{k>=1} (1 + x^(4*k-2)) / ((1 + x^(4*k-1)) * (1 - x^(4*k-3))).
a(n) ~ Gamma(1/4) * exp(Pi*sqrt(n/6)) / (4 * 6^(1/8) * Pi^(3/4) * n^(5/8)). - Vaclav Kotesovec, Jul 17 2019

cp's OEIS Frontend

A284316 Expansion of Product_{k>=0} (1 - x^(4*k+3)) in powers of x.

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A301505 Expansion of Product_{k>=1} (1 + x^(4*k))*(1 + x^(4*k-1)).

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A301508 Expansion of Product_{k>=0} (1 + x^(4*k+2))*(1 + x^(4*k+3)).

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

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A170960 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 7.

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A170961 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 8.

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A170962 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 9.

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A170963 Expansion of Product_{i=1..m} (1 + x^(4*i-1)) for m = 10.

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A301505 Expansion of Product_{k>=1} (1 + x^(4k))(1 + x^(4*k-1)).

A301508 Expansion of Product_{k>=0} (1 + x^(4k+2))(1 + x^(4*k+3)).

A309240 Expansion of 1/((1 - x)(1 - x^2)(1 + x^3)(1 + x^4)(1 - x^5)(1 - x^6)(1 + x^7)(1 + x^8)...).