A000335 -id:A000335 - cp's OEIS frontend

A338046 G.f.: Sum_{k>=0} x^(2^k) / (1 - x^(2^k))^4.

1, 5, 10, 25, 35, 66, 84, 145, 165, 255, 286, 430, 455, 644, 680, 961, 969, 1305, 1330, 1795, 1771, 2310, 2300, 3030, 2925, 3731, 3654, 4704, 4495, 5640, 5456, 6945, 6545, 8109, 7770, 9741, 9139, 11210, 10660, 13275, 12341, 15015, 14190, 17490, 16215, 19596, 18424, 22630

Offset: 1

Ilya Gutkovskiy, Oct 08 2020

nonn

Cf. A000292, A000335, A000447, A001511, A002492, A129527, A209229, A328407, A338045.

nmax = 48; CoefficientList[Series[Sum[x^(2^k) /(1 - x^(2^k))^4, {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x] // Rest
a[n_] := If[EvenQ[n], a[n/2] + n (n + 1) (n + 2)/6, n (n + 1) (n + 2)/6]; Table[a[n], {n, 1, 48}]
Table[(1/6) DivisorSum[n, Boole[IntegerQ[Log[2, n/#]]] # (# + 1) (# + 2) &], {n, 1, 48}]

G.f. A(x) satisfies: A(x) = A(x^2) + x / (1 - x)^4.
a(2*n) = a(n) + A002492(n), a(2*n+1) = A000447(n+1).
a(n) = (1/6) * Sum_{d|n} A209229(n/d) * d * (d + 1) * (d + 2).
Product_{n>=1} (1 + x^n)^a(n) = g.f. for A000335.

A279761 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(2k^2+1)/3).

Original entry on oeis.org

1, 1, 7, 26, 91, 290, 946, 2922, 8937, 26521, 77485, 222005, 626988, 1743739, 4787625, 12979799, 34792728, 92257673, 242197348, 629805075, 1623197726, 4148192991, 10516418844, 26458470616, 66086152465, 163925621199, 403931474096, 989040788801, 2407020523315, 5823830868091, 14011949899801, 33530477120905, 79820957945103

Offset: 0

Ilya Gutkovskiy, Dec 18 2016

nonn

Euler transform of the octahedral numbers (A005900).

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Octahedral Number
OEIS Wiki, Platonic numbers

Cf. A000335, A005900, A023872.

nmax=32; CoefficientList[Series[Product[1/(1 - x^k)^(k (2 k^2 + 1)/3), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^k)^(k*(2*k^2+1)/3).

a(n) ~ exp(Zeta'(-1)/3 - Zeta(3)^2 / (360*Zeta(5)) + 2*Zeta'(-3)/3 + (Zeta(3)/(6*2^(3/5) * Zeta(5)^(2/5))) * n^(2/5) + (5*(Zeta(5)/2)^(1/5)/2) * n^(4/5)) * Zeta(5)^(47/450) / (2^(37/450) * sqrt(5*Pi) * n^(136/225)). - Vaclav Kotesovec, Nov 09 2017

A279762 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(5k^2-5*k+2)/2).

Original entry on oeis.org

1, 1, 13, 61, 263, 1094, 4578, 18076, 69815, 262242, 965342, 3480006, 12322360, 42896002, 147062818, 497000146, 1657470977, 5459160063, 17772284155, 57225458626, 182362100816, 575463112191, 1799106136923, 5575063264825, 17130798464652, 52216240087807, 157937816918539, 474197830869573, 1413695306175884, 4185962563381518

Offset: 0

Ilya Gutkovskiy, Dec 18 2016

nonn

Euler transform of the icosahedral numbers (A006564).

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
OEIS Wiki, Platonic numbers

Cf. A000335, A006564, A023872.

nmax=29; CoefficientList[Series[Product[1/(1 - x^k)^(k (5 k^2 - 5 k + 2)/2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^k)^(k*(5*k^2-5*k+2)/2).

a(n) ~ exp(Zeta'(-1) + 5*Zeta(3) / (8*Pi^2) - Pi^16 / (16796160000*Zeta(5)^3) + Pi^8 * Zeta(3) / (648000*Zeta(5)^2) - Zeta(3)^2 / (150*Zeta(5)) + 5*Zeta'(-3)/2 + (-Pi^12/(19440000 * 2^(2/5) * 15^(1/5) * Zeta(5)^(11/5)) + Pi^4 * Zeta(3) / (900 * 2^(2/5) * 15^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8 / (21600 * 2^(4/5) * 15^(2/5) * Zeta(5)^(7/5)) + Zeta(3) / (2^(4/5) * (15*Zeta(5))^(2/5))) * n^(2/5) + (-Pi^4 / (36 * 2^(1/5) * (15*Zeta(5))^(3/5))) * n^(3/5) + ((5*(15*Zeta(5))^(1/5)) / 2^(8/5)) * n^(4/5)) * (3*Zeta(5))^(9/80) / (2^(11/40) * 5^(31/80) * sqrt(Pi) * n^(49/80)). - Vaclav Kotesovec, Nov 09 2017

A279763 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(3k-1)(3k-2)/2).

Original entry on oeis.org

1, 1, 21, 105, 535, 2670, 12996, 59546, 266875, 1161894, 4939778, 20528320, 83636061, 334496221, 1315381029, 5091782355, 19424086781, 73092029218, 271537720562, 996656173345, 3616680935702, 12983391870459, 46133749660407, 162337625047433, 565962994479384, 1955721907216420, 6701061533668542, 22774651422340672

Offset: 0

Ilya Gutkovskiy, Dec 18 2016

nonn

Euler transform of the dodecahedral numbers (A006566).

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
OEIS Wiki, Platonic numbers

Cf. A000335, A006566, A023872.

nmax=27; CoefficientList[Series[Product[1/(1 - x^k)^(k (3 k - 1) (3 k - 2)/2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^k)^(k*(3*k-1)*(3*k-2)/2).

a(n) ~ exp(Zeta'(-1) + 9*Zeta(3) / (8*Pi^2) - Pi^16 / (9331200000*Zeta(5)^3) + Pi^8 * Zeta(3) / (648000*Zeta(5)^2) - Zeta(3)^2 / (270*Zeta(5)) + 9*Zeta'(-3)/2 + (-Pi^12/(10800000 * 2^(2/5) * 3^(3/5) * Zeta(5)^(11/5)) + Pi^4 * Zeta(3) / (900 * 2^(2/5) * 3^(3/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8 / (36000 * 2^(4/5) * 3^(1/5) * Zeta(5)^(7/5)) + Zeta(3) / (2^(4/5) * 3^(6/5) * Zeta(5)^(2/5))) * n^(2/5) + (-Pi^4 / (60 * 2^(1/5) * 3^(4/5) * Zeta(5)^(3/5))) * n^(3/5) + ((5*3^(3/5) * Zeta(5)^(1/5)) / 2^(8/5)) * n^(4/5)) * 3^(131/400) * Zeta(5)^(131/1200) / (2^(169/600) * sqrt(5*Pi) * n^(731/1200)). - Vaclav Kotesovec, Nov 09 2017

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A338046 G.f.: Sum_{k>=0} x^(2^k) / (1 - x^(2^k))^4.

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A279761 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(2k^2+1)/3).

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A279762 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(5k^2-5*k+2)/2).

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A279763 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(3k-1)(3k-2)/2).

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cp's OEIS Frontend

A338046 G.f.: Sum_{k>=0} x^(2^k) / (1 - x^(2^k))^4.

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A279761 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(2*k^2+1)/3).

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A279762 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(5*k^2-5*k+2)/2).

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A279763 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(3*k-1)*(3*k-2)/2).

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A279761 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(2k^2+1)/3).

A279762 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(5k^2-5*k+2)/2).

A279763 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(3k-1)(3k-2)/2).