A169975 -id:A169975 - cp's OEIS frontend

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

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

A284313 Expansion of Product_{k>=0} (1 - x^(4*k+1)) in powers of x.

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Mar 24 2017

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

Cf. Product_{k>=0} (1 - x^(m*k+1)): A081362 (m=2), A284312 (m=3), this sequence (m=4), A284314 (m=5).

Cf. A050449, A169975, A284316.

V:= Vector(100):
V[1]:= 1:
for k from 0 to 24 do
  V[4*k+2..100]:= V[4*k+2..100] - V[1..99-4*k]
od:
convert(V,list); # Robert Israel, May 03 2017

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

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

a(n) = -(1/n)*Sum_{k=1..n} A050449(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. A284316. - Peter Bala, Nov 28 2020

A339059 Number of compositions (ordered partitions) of n into distinct parts congruent to 1 mod 4.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 2, 0, 0, 1, 2, 0, 0, 1, 4, 6, 0, 1, 4, 6, 0, 1, 6, 12, 0, 1, 6, 18, 24, 1, 8, 24, 24, 1, 8, 30, 48, 1, 10, 42, 72, 1, 10, 48, 120, 121, 12, 60, 144, 121, 12, 72, 216, 241, 14, 84, 264, 361, 14, 96, 360, 601, 16, 114, 432, 841, 736, 126, 552, 1201, 738

Offset: 0

Ilya Gutkovskiy, Nov 22 2020

nonn

			a(15) = 6 because we have [9, 5, 1], [9, 1, 5], [5, 9, 1], [5, 1, 9], [1, 9, 5] and [1, 5, 9].

Index entries for sequences related to compositions

Cf. A003269, A016813, A032020, A032021, A035451, A169975, A337547, A337548, A339060.

nmax = 70; CoefficientList[Series[Sum[k! x^(k (2 k - 1))/Product[1 - x^(4 j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=0} k! * x^(k*(2*k - 1)) / Product_{j=1..k} (1 - x^(4*j)).

A261634 Expansion of Product_{k>=0} (1+x^(4*k+1))^3.

Original entry on oeis.org

1, 3, 3, 1, 0, 3, 9, 9, 3, 3, 12, 18, 12, 6, 18, 37, 33, 15, 22, 54, 66, 42, 36, 81, 114, 84, 57, 112, 189, 171, 109, 156, 279, 294, 201, 222, 405, 486, 360, 328, 564, 747, 617, 504, 783, 1123, 1017, 783, 1065, 1602, 1605

Offset: 0

Vaclav Kotesovec, Aug 27 2015

nonn

Cf. A261612, A261615, A261633, A169975, A261630.

nmax=50; CoefficientList[Series[Product[(1+x^(4*k+1))^3, {k, 0, nmax}], {x, 0, nmax}], x]

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

A301507 Expansion of Product_{k>=0} (1 + x^(4k+1))(1 + x^(4*k+2)).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4, 5, 6, 6, 6, 7, 8, 9, 10, 11, 13, 14, 14, 16, 18, 20, 23, 24, 27, 30, 31, 34, 37, 41, 46, 49, 53, 58, 62, 67, 73, 80, 88, 94, 101, 109, 117, 127, 136, 147, 161, 172, 184, 198, 211, 228, 245, 262, 284, 304, 324, 347, 370, 397, 425, 454, 488

Offset: 0

Ilya Gutkovskiy, Mar 22 2018

nonn

Number of partitions of n into distinct parts congruent to 1 or 2 mod 4.

			a(11) = 3 because we have [10, 1], [9, 2] and [6, 5].

Index entries for sequences related to partitions

Cf. A035365, A042963, A070048, A108932, A169975, A301504, A301505, A301508.

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

G.f.: Product_{k>=1} (1 + x^A042963(k)).

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

A170975 Expansion of Product_{i=0..m-1} (1 + x^(4*i+1)) for m = 12.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 3, 2, 0, 1, 3, 3, 1, 1, 4, 4, 1, 1, 4, 5, 2, 1, 5, 7, 3, 1, 5, 8, 5, 2, 6, 10, 6, 1, 5, 12, 9, 2, 5, 13, 11, 3, 4, 14, 15, 5, 4, 15, 17, 7, 4, 15, 21, 10, 4, 15, 23, 13, 4, 15, 27, 17, 5, 14, 28, 21, 6, 13, 31, 26, 8, 12, 31, 30, 11, 11

Offset: 0

N. J. A. Sloane, Aug 29 2010

Nathaniel Johnston, Table of n, a(n) for n = 0..276 (full sequence)

Cf. A169975, A170966-A170984.

m:=12; R:=PowerSeriesRing(Integers(), 100); Coefficients(R!( (&*[1+x^(4*j+1): j in [0..m-1]]) )); // G. C. Greubel, Feb 24 2019

seq(coeff(mul((1+x^(4*i+1)),i=0..11),x,n),n=0..100); # Nathaniel Johnston, Jun 24 2011

With[{m=12}, CoefficientList[Series[Product[(1 + x^(4*j+1)), {j,0,m-1}], {x,0,100}],x]] (* G. C. Greubel, Feb 24 2019 *)

m=12; my(x='x+O('x^(100))); Vec(prod(j=0,m-1, 1+x^(4*j+1) )) \\ G. C. Greubel, Feb 24 2019

m=12; ( prod(1+x^(4*j+1) for j in (0..m-1)) ).series(x, 100).coefficients(x, sparse=False) # G. C. Greubel, Feb 24 2019

a(n) = a(276-n). - Rick L. Shepherd, Mar 01 2013

Typo in Maple program fixed and b-file extended 9 terms by Rick L. Shepherd, Mar 01 2013

A261630 Expansion of Product_{k>=0} (1+x^(4*k+1))^2.

Original entry on oeis.org

1, 2, 1, 0, 0, 2, 4, 2, 0, 2, 5, 4, 1, 2, 8, 10, 4, 2, 9, 14, 9, 4, 12, 22, 16, 6, 13, 30, 30, 14, 17, 40, 44, 24, 21, 50, 66, 42, 29, 64, 92, 66, 41, 78, 127, 106, 62, 96, 164, 152, 93, 120, 215, 220, 139, 150, 269, 302, 205, 192, 340, 412, 296, 248, 417

Offset: 0

Vaclav Kotesovec, Aug 27 2015

nonn

Cf. A169975, A261634, A261638.

nmax=80; CoefficientList[Series[Product[(1+x^(4*k+1))^2, {k, 0, nmax}], {x, 0, nmax}], x]

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

A261638 Expansion of Product_{k>=0} (1+x^(4*k+1))^4.

Original entry on oeis.org

1, 4, 6, 4, 1, 4, 16, 24, 16, 8, 22, 48, 52, 32, 38, 92, 128, 96, 70, 140, 245, 244, 172, 228, 417, 488, 374, 380, 680, 924, 798, 676, 1044, 1560, 1542, 1256, 1625, 2524, 2778, 2304, 2537, 3892, 4716, 4156, 4076, 5908, 7650, 7196, 6592, 8796, 11938

Offset: 0

Vaclav Kotesovec, Aug 27 2015

nonn

In general, if j > 0, a > 0, b > 0, GCD(a,b) = 1 and g.f. = Product_{k>=0} (1 + x^(a*k+b))^j, then a(n) ~ 2^((j-3)/2 - j*b/a) * j^(1/4) * exp(Pi*sqrt(j*n/(3*a))) / ((3*a)^(1/4) * n^(3/4)).

Cf. A261612, A261615, A261637, A169975, A261630, A261634.

nmax=50; CoefficientList[Series[Product[(1+x^(4*k+1))^4, {k, 0, nmax}], {x, 0, nmax}], x]

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

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

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

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

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A284313 Expansion of Product_{k>=0} (1 - x^(4*k+1)) in powers of x.

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A339059 Number of compositions (ordered partitions) of n into distinct parts congruent to 1 mod 4.

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A261634 Expansion of Product_{k>=0} (1+x^(4*k+1))^3.

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A301507 Expansion of Product_{k>=0} (1 + x^(4*k+1))*(1 + x^(4*k+2)).

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A170975 Expansion of Product_{i=0..m-1} (1 + x^(4*i+1)) for m = 12.

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A261630 Expansion of Product_{k>=0} (1+x^(4*k+1))^2.

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A261638 Expansion of Product_{k>=0} (1+x^(4*k+1))^4.

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A301507 Expansion of Product_{k>=0} (1 + x^(4k+1))(1 + x^(4*k+2)).