A301569 -id:A301569

A301562

Expansion of Product_{k>=0} (1 + x^(5*k+1))*(1 + x^(5*k+2)).

+10
7

1, 1, 1, 1, 0, 0, 1, 2, 2, 2, 1, 1, 2, 3, 3, 2, 2, 3, 5, 6, 5, 4, 4, 6, 8, 8, 7, 7, 9, 12, 13, 11, 10, 12, 16, 19, 19, 17, 18, 23, 27, 27, 25, 25, 30, 37, 40, 38, 37, 42, 50, 55, 54, 52, 57, 68, 77, 78, 75, 78, 90, 102, 106, 104, 106, 120, 138, 146, 144, 145, 158

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,8

COMMENTS

Number of partitions of n into distinct parts congruent to 1 or 2 mod 5.

LINKS

Table of n, a(n) for n=0..70.

Index entries for sequences related to partitions

FORMULA

G.f.: Product_{k>=1} (1 + x^A047216(k)).

a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(17/20) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 24 2018

EXAMPLE

a(13) = 3 because we have [12, 1], [11, 2] and [7, 6].

MATHEMATICA

nmax = 70; CoefficientList[Series[Product[(1 + x^(5 k + 1)) (1 + x^(5 k + 2)), {k, 0, nmax}], {x, 0, nmax}], x]

nmax = 70; CoefficientList[Series[QPochhammer[-x, x^5] QPochhammer[-x^2, x^5], {x, 0, nmax}], x]

nmax = 70; CoefficientList[Series[Product[(1 + Boole[MemberQ[{1, 2}, Mod[k, 5]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

CROSSREFS

Cf. A035371, A047216, A107234, A107235, A203776, A219607, A280454, A281272, A301563, A301564, A301565, A301567, A301568, A301569, A301570.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Mar 23 2018

STATUS

approved

A301563

Expansion of Product_{k>=0} (1 + x^(5*k+1))*(1 + x^(5*k+3)).

+10
7

1, 1, 0, 1, 1, 0, 1, 1, 1, 2, 1, 2, 2, 1, 3, 2, 2, 4, 3, 4, 4, 4, 6, 4, 6, 7, 5, 9, 8, 8, 11, 9, 12, 12, 12, 16, 13, 17, 19, 17, 23, 21, 24, 27, 24, 32, 30, 32, 40, 35, 43, 45, 44, 53, 50, 59, 62, 61, 75, 70, 78, 87, 83, 99, 97, 105, 118, 112, 133, 134, 138, 159, 153

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,10

COMMENTS

Number of partitions of n into distinct parts congruent to 1 or 3 mod 5.

LINKS

Table of n, a(n) for n=0..72.

Index entries for sequences related to partitions

FORMULA

G.f.: Product_{k>=1} (1 + x^A047219(k)).

a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(21/20) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 24 2018

EXAMPLE

a(14) = 3 because we have [13, 1], [11, 3] and [8, 6].

MATHEMATICA

nmax = 72; CoefficientList[Series[Product[(1 + x^(5 k + 1)) (1 + x^(5 k + 3)), {k, 0, nmax}], {x, 0, nmax}], x]

nmax = 72; CoefficientList[Series[QPochhammer[-x, x^5] QPochhammer[-x^3, x^5], {x, 0, nmax}], x]

nmax = 72; CoefficientList[Series[Product[(1 + Boole[MemberQ[{1, 3}, Mod[k, 5]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

CROSSREFS

Cf. A035372, A047219, A107234, A107236, A203776, A219607, A280454, A281271, A301562, A301564, A301565, A301567, A301568, A301569, A301570.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Mar 23 2018

STATUS

approved

A301564

Expansion of Product_{k>=0} (1 + x^(5*k+2))*(1 + x^(5*k+4)).

+10
7

1, 0, 1, 0, 1, 0, 1, 1, 0, 2, 0, 2, 1, 2, 2, 1, 3, 1, 3, 3, 2, 5, 2, 6, 3, 5, 6, 4, 8, 5, 8, 8, 7, 12, 7, 13, 11, 11, 16, 11, 19, 14, 19, 21, 17, 27, 20, 27, 28, 26, 36, 28, 40, 37, 38, 49, 39, 55, 49, 55, 64, 55, 76, 65, 78, 84, 78, 100, 87, 107, 109, 107, 134, 116, 145

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,10

COMMENTS

Number of partitions of n into distinct parts congruent to 2 or 4 mod 5.

LINKS

Table of n, a(n) for n=0..74.

Index entries for sequences related to partitions

FORMULA

G.f.: Product_{k>=1} (1 + x^A047211(k)).

a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(29/20) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 24 2018

EXAMPLE

a(16) = 3 because we have [14, 2], [12, 4] and [9, 7].

MATHEMATICA

nmax = 74; CoefficientList[Series[Product[(1 + x^(5 k + 2)) (1 + x^(5 k + 4)), {k, 0, nmax}], {x, 0, nmax}], x]

nmax = 74; CoefficientList[Series[QPochhammer[-x^2, x^5] QPochhammer[-x^4, x^5], {x, 0, nmax}], x]

nmax = 74; CoefficientList[Series[Product[(1 + Boole[MemberQ[{2, 4}, Mod[k, 5]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

CROSSREFS

Cf. A035373, A047211, A107235, A107237, A203776, A219607, A281243, A281272, A301562, A301563, A301565, A301567, A301568, A301569, A301570.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Mar 23 2018

STATUS

approved

A301565

Expansion of Product_{k>=0} (1 + x^(5*k+3))*(1 + x^(5*k+4)).

+10
7

1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 2, 2, 1, 1, 2, 3, 2, 1, 2, 4, 4, 3, 3, 4, 6, 6, 4, 4, 7, 9, 7, 6, 8, 11, 12, 10, 9, 12, 16, 16, 14, 14, 19, 23, 22, 19, 21, 27, 31, 29, 26, 31, 40, 42, 38, 38, 45, 53, 55, 51, 52, 63, 73, 73, 69, 73, 87, 97, 95, 91, 100, 118, 128

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,13

COMMENTS

Number of partitions of n into distinct parts congruent to 3 or 4 mod 5.

LINKS

Table of n, a(n) for n=0..74.

Index entries for sequences related to partitions

FORMULA

G.f.: Product_{k>=1} (1 + x^A047204(k)).

a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(33/20) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 24 2018

EXAMPLE

a(17) = 3 because we have [14, 3], [13, 4] and [9, 8].

MATHEMATICA

nmax = 74; CoefficientList[Series[Product[(1 + x^(5 k + 3)) (1 + x^(5 k + 4)), {k, 0, nmax}], {x, 0, nmax}], x]

nmax = 74; CoefficientList[Series[QPochhammer[-x^3, x^5] QPochhammer[-x^4, x^5], {x, 0, nmax}], x]

nmax = 74; CoefficientList[Series[Product[(1 + Boole[MemberQ[{3, 4}, Mod[k, 5]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

CROSSREFS

Cf. A035374, A047204, A107236, A107237, A203776, A219607, A281243, A281271, A301562, A301563, A301564, A301567, A301568, A301569, A301570.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Mar 23 2018

STATUS

approved

A301567

Expansion of Product_{k>=1} (1 + x^(5*k))*(1 + x^(5*k-4)).

+10
7

1, 1, 0, 0, 0, 1, 2, 1, 0, 0, 1, 3, 2, 0, 0, 2, 5, 4, 1, 0, 2, 7, 7, 2, 0, 3, 10, 11, 4, 0, 4, 14, 17, 8, 1, 5, 19, 25, 13, 2, 6, 25, 36, 21, 4, 8, 33, 50, 33, 8, 10, 43, 69, 49, 14, 13, 55, 93, 71, 23, 17, 70, 124, 102, 37, 22, 88, 163, 142, 57, 30, 110, 212, 195, 85

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,7

COMMENTS

Number of partitions of n into distinct parts congruent to 0 or 1 mod 5.

LINKS

Table of n, a(n) for n=0..74.

Index entries for sequences related to partitions

FORMULA

G.f.: Product_{k>=2} (1 + x^A008851(k)).

a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(29/20) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 24 2018

EXAMPLE

a(11) = 3 because we have [11], [10, 1] and [6, 5].

MATHEMATICA

nmax = 74; CoefficientList[Series[Product[(1 + x^(5 k)) (1 + x^(5 k - 4)), {k, 1, nmax}], {x, 0, nmax}], x]

nmax = 74; CoefficientList[Series[x^4 QPochhammer[-1, x^5] QPochhammer[-x^(-4), x^5]/(2 (1 + x^4)), {x, 0, nmax}], x]

nmax = 74; CoefficientList[Series[Product[(1 + Boole[MemberQ[{0, 1}, Mod[k, 5]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

CROSSREFS

Cf. A008851, A035367, A036820, A203776, A219607, A280454, A301562, A301563, A301564, A301565, A301568, A301569, A301570.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Mar 23 2018

STATUS

approved

A301568

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

+10
7

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 1, 0, 3, 0, 2, 2, 0, 5, 0, 4, 2, 1, 7, 0, 7, 3, 2, 10, 0, 11, 4, 4, 14, 0, 17, 5, 8, 19, 1, 25, 6, 13, 25, 2, 36, 8, 21, 33, 4, 50, 10, 33, 43, 8, 69, 12, 49, 55, 14, 93, 16, 71, 70, 23, 124, 20, 102, 88, 37, 163, 26, 142, 110, 57, 212

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,8

COMMENTS

Number of partitions of n into distinct parts congruent to 0 or 2 mod 5.

LINKS

Table of n, a(n) for n=0..74.

Index entries for sequences related to partitions

FORMULA

G.f.: Product_{k>=1} (1 + x^A047215(k)).

a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(33/20) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 24 2018

EXAMPLE

a(12) = 3 because we have [12], [10, 2] and [7, 5].

MATHEMATICA

nmax = 74; CoefficientList[Series[Product[(1 + x^(5 k)) (1 + x^(5 k - 3)), {k, 1, nmax}], {x, 0, nmax}], x]

nmax = 74; CoefficientList[Series[x^3 QPochhammer[-1, x^5] QPochhammer[-x^(-3), x^5]/(2 (1 + x) (1 - x + x^2)), {x, 0, nmax}], x]

nmax = 74; CoefficientList[Series[Product[(1 + Boole[MemberQ[{0, 2}, Mod[k, 5]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

CROSSREFS

Cf. A035368, A036822, A047215, A203776, A219607, A281272, A301562, A301563, A301564, A301565, A301567, A301569, A301570.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Mar 23 2018

STATUS

approved

A301570

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

+10
7

1, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 1, 3, 2, 0, 0, 2, 5, 2, 0, 0, 4, 7, 3, 0, 1, 7, 10, 4, 0, 2, 11, 14, 5, 0, 4, 17, 19, 6, 0, 8, 25, 25, 8, 1, 13, 36, 33, 10, 2, 21, 50, 43, 12, 4, 33, 69, 55, 15, 8, 49, 93, 70, 18, 14, 71, 124, 88, 23, 23, 102, 163, 110, 29, 37

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,10

COMMENTS

Number of partitions of n into distinct parts congruent to 0 or 4 mod 5.

LINKS

Table of n, a(n) for n=0..76.

Index entries for sequences related to partitions

FORMULA

G.f.: Product_{k>=2} (1 + x^A047208(k)).

a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(41/20) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 24 2018

EXAMPLE

a(14) = 3 because we have [14], [10, 4] and [9, 5].

MATHEMATICA

nmax = 76; CoefficientList[Series[Product[(1 + x^(5 k)) (1 + x^(5 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]

nmax = 76; CoefficientList[Series[x QPochhammer[-1, x^5] QPochhammer[-x^(-1), x^5]/(2 (1 + x)), {x, 0, nmax}], x]

nmax = 76; CoefficientList[Series[Product[(1 + Boole[MemberQ[{0, 4}, Mod[k, 5]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

CROSSREFS

Cf. A035370, A036820, A047208, A203776, A219607, A281243, A301562, A301563, A301564, A301565, A301567, A301568, A301569.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Mar 23 2018

STATUS

approved