A035451 -id:A035451 - cp's OEIS frontend

A301504 Expansion of Product_{k>=1} (1 + x^(4k))(1 + x^(4*k-3)).

1, 1, 0, 0, 1, 2, 1, 0, 1, 3, 2, 0, 2, 5, 4, 1, 2, 7, 7, 2, 3, 10, 11, 4, 4, 14, 17, 8, 6, 19, 25, 13, 8, 25, 36, 21, 12, 33, 50, 33, 18, 43, 69, 49, 26, 56, 93, 71, 38, 72, 124, 102, 55, 92, 163, 142, 79, 118, 212, 195, 112, 151, 273, 265, 157, 193, 350, 354, 217, 246, 444

Offset: 0

Ilya Gutkovskiy, Mar 22 2018

nonn

Number of partitions of n into distinct parts congruent to 0 or 1 mod 4.

			a(9) = 3 because we have [9], [8, 1] and [5, 4].

Index entries for sequences related to partitions

Cf. A035362, A035451, A035457, A042948, A108932, A132463, A261734, A301505, A301507, A301508.

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

G.f.: Product_{k>=1} (1 + x^A042948(k)).

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

A117957 Number of partitions of n into parts larger than 1 and congruent to 1 mod 4.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 2, 1, 1, 1, 2, 2, 1, 2, 3, 3, 2, 2, 4, 4, 3, 3, 5, 6, 5, 4, 6, 8, 7, 6, 8, 10, 10, 9, 10, 13, 13, 12, 14, 17, 18, 16, 18, 22, 23, 22, 23, 28, 31, 29, 30, 36, 39, 39, 39, 45, 51, 50, 51, 57, 64, 65, 65, 73, 81, 83, 84, 91, 102, 106, 106

Offset: 0

Emeric Deutsch, Apr 05 2006

nonn

Also number of partitions of n such that 2k and 2k+1 occur with the same multiplicities. Example: a(26)=3 because we have [11,10,3,2], [9,8,5,4] and [7,7,6,6]. It is easy to find a bijection between these partitions and those described in the definition.

			a(26)=3 because we have [21,5],[17,9] and [13,13].

Cf. A035451, A035462.

g:=1/product(1-x^(4*i+1),i=1..50): gser:=series(g,x=0,93): seq(coeff(gser,x,n),n=0..88);

nmax = 100; CoefficientList[Series[Product[1/(1-x^(4*k+1)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 07 2016 *)

G.f.: 1/product(1-x^(4i+1), i=1..infinity).

a(n) ~ exp(sqrt(n/6)*Pi) * Pi^(1/4) * Gamma(1/4) / (2^(31/8) * 3^(5/8) * n^(9/8)). - Vaclav Kotesovec, Mar 07 2016

A261629 Expansion of Product_{k>=0} 1/(1-x^(4*k+1))^2.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 11, 14, 17, 22, 30, 38, 46, 56, 70, 88, 106, 126, 153, 186, 224, 264, 312, 372, 440, 516, 603, 708, 830, 964, 1117, 1296, 1503, 1734, 1992, 2292, 2638, 3024, 3453, 3942, 4504, 5134, 5831, 6616, 7511, 8518, 9631, 10872, 12274, 13848, 15592

Offset: 0

Vaclav Kotesovec, Aug 27 2015

nonn

Cf. A035451, A261632, A261636.

nmax=60; CoefficientList[Series[Product[1/(1-x^(4*k+1))^2, {k, 0, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt(n/3)) * Gamma(1/4)^2 / (8 * Pi^(3/2) * sqrt(n)).

A261632 Expansion of Product_{k>=0} 1/(1-x^(4*k+1))^3.

Original entry on oeis.org

1, 3, 6, 10, 15, 24, 37, 54, 75, 103, 144, 198, 265, 348, 456, 599, 777, 993, 1262, 1602, 2028, 2543, 3165, 3930, 4868, 6003, 7359, 8991, 10965, 13329, 16138, 19473, 23448, 28171, 33738, 40293, 48025, 57132, 67803, 80267, 94845, 111888, 131736, 154779, 181530

Offset: 0

Vaclav Kotesovec, Aug 27 2015

nonn

Cf. A035382, A261616, A261631, A035451, A261629.

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

a(n) ~ exp(Pi*sqrt(n/2)) * 2^(3/8) * Gamma(1/4)^3 / (16 * Pi^(9/4) * n^(3/8)).

A261636 Expansion of Product_{k>=0} 1/(1-x^(4*k+1))^4.

Original entry on oeis.org

1, 4, 10, 20, 35, 60, 100, 160, 245, 364, 536, 780, 1115, 1564, 2166, 2980, 4065, 5484, 7326, 9720, 12830, 16824, 21902, 28344, 36510, 46820, 59736, 75844, 95910, 120844, 151688, 189668, 236330, 293564, 363542, 448804, 552425, 678144, 830338, 1014052, 1235296

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/(1 - x^(a*k+b))^j, then a(n) ~ Gamma(b/a)^j * 2^(-(j+5)/4 - j*b/(2*a)) * 3^((j-1)/4 - j*b/(2*a)) * j^(-(j-1)/4 + j*b/(2*a)) * a^(-(j+1)/4 + j*b/(2*a)) * Pi^(-j + j*b/a) * n^((j-3)/4 - j*b/(2*a)) * exp(Pi*sqrt(2*j*n/(3*a))).

Cf. A035382, A261616, A261635, A035451, A261629, A261632.

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

a(n) ~ exp(Pi*sqrt(2*n/3)) * 6^(1/4) * Gamma(1/4)^4 / (32 * Pi^3 * n^(1/4)).

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

A301504 Expansion of Product_{k>=1} (1 + x^(4*k))*(1 + x^(4*k-3)).

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A117957 Number of partitions of n into parts larger than 1 and congruent to 1 mod 4.

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A261629 Expansion of Product_{k>=0} 1/(1-x^(4*k+1))^2.

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A261632 Expansion of Product_{k>=0} 1/(1-x^(4*k+1))^3.

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A261636 Expansion of Product_{k>=0} 1/(1-x^(4*k+1))^4.

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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)*...).

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A301504 Expansion of Product_{k>=1} (1 + x^(4k))(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)...).