A279218 -id:A279218 - cp's OEIS frontend

A279215 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(k+1)(2*k+1)/6).

1, 1, 6, 20, 65, 190, 571, 1616, 4555, 12439, 33515, 88517, 230738, 592321, 1502384, 3763946, 9328899, 22880511, 55585077, 133806273, 319373068, 756124040, 1776497540, 4143489680, 9597505006, 22083821765, 50494638926, 114758996621, 259303832735, 582655202940, 1302234303910, 2895530963661, 6406348746390

Offset: 0

Ilya Gutkovskiy, Dec 08 2016

nonn

Euler transform of the square pyramidal numbers (A000330).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
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, Square Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A000330, A000335, A279216, A279217, A279218, A279219.

nmax=32; CoefficientList[Series[Product[1/(1 - x^k)^(k (k + 1) (2 k + 1)/6), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^k)^(k*(k+1)*(2*k+1)/6).

a(n) ~ exp(Zeta'(-1)/6 - Zeta(3)/(8*Pi^2) - Pi^16/(24883200000*Zeta(5)^3) + Pi^8*Zeta(3)/(1728000*Zeta(5)^2) - Zeta(3)^2/(720*Zeta(5)) + Zeta'(-3)/3 + (Pi^12/(43200000*2^(3/5)*Zeta(5)^(11/5)) - Pi^4*Zeta(3) / (3600*2^(3/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8/(144000*2^(1/5)*Zeta(5)^(7/5)) + Zeta(3)/(12*2^(1/5)*Zeta(5)^(2/5))) * n^(2/5) + Pi^4/(180*2^(4/5)*Zeta(5)^(3/5)) * n^(3/5) + 5*Zeta(5)^(1/5)/2^(7/5) * n^(4/5)) * Zeta(5)^(23/225) / (2^(29/150) * sqrt(5*Pi) * n^(271/450)). - Vaclav Kotesovec, Dec 08 2016

A279216 Expansion of Product_{k>=1} 1/(1 - x^k)^(k^2*(k+1)/2).

Original entry on oeis.org

1, 1, 7, 25, 86, 269, 862, 2606, 7812, 22704, 64989, 182356, 504414, 1373694, 3693367, 9804435, 25733084, 66808578, 171719539, 437183839, 1103143657, 2760037810, 6850400668, 16873338215, 41260373472, 100196920196, 241712863504, 579416535973, 1380517695672, 3270075208145, 7702580246941

Offset: 0

Ilya Gutkovskiy, Dec 08 2016

nonn

Euler transform of the pentagonal pyramidal numbers (A002411).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
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, Pentagonal Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A000335, A002411, A279215, A279217, A279218, A279219.

nmax=30; CoefficientList[Series[Product[1/(1 - x^k)^(k^2 (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

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

a(n) ~ exp(-Zeta(3)/(8*Pi^2) - Pi^16/(83980800000*Zeta(5)^3) + Zeta'(-3)/2 + (Pi^12/(97200000*2^(2/5)*3^(1/5)*Zeta(5)^(11/5))) * n^(1/5) + (-Pi^8/(108000*2^(4/5)*3^(2/5)*Zeta(5)^(7/5))) * n^(2/5) + (Pi^4/(180*2^(1/5)*(3*Zeta(5))^(3/5))) * n^(3/5) + ((5*(3*Zeta(5))^(1/5))/(2^(8/5))) * n^(4/5)) * (3*Zeta(5))^(119/1200) / (2^(181/600) * sqrt(5*Pi) * n^(719/1200)). - Vaclav Kotesovec, Dec 08 2016

A279217 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(k+1)(4*k-1)/6).

Original entry on oeis.org

1, 1, 8, 30, 108, 357, 1205, 3838, 12083, 36896, 110828, 326281, 946086, 2700026, 7602642, 21128513, 58028309, 157588912, 423534324, 1127102360, 2971764946, 7766890826, 20131080168, 51766513279, 132117237595, 334770353022, 842462217948, 2106183375971, 5232414548275, 12920429411759, 31719180847831

Offset: 0

Ilya Gutkovskiy, Dec 08 2016

nonn

Euler transform of the hexagonal pyramidal numbers (A002412).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
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, Hexagonal Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A000335, A002412, A279215, A279216, A279218, A279219.

nmax=30; CoefficientList[Series[Product[1/(1 - x^k)^(k (k + 1)(4 k - 1)/6), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^k)^(k*(k+1)*(4*k-1)/6).

a(n) ~ exp(-Zeta'(-1)/6 - Zeta(3)/(8*Pi^2) - Pi^16/(199065600000*Zeta(5)^3) - Pi^8*Zeta(3)/(6912000*Zeta(5)^2) - Zeta(3)^2/(1440*Zeta(5)) + 2*Zeta'(-3)/3 + (Pi^12/(172800000*2^(4/5)*Zeta(5)^(11/5)) + Pi^4*Zeta(3)/(7200*2^(4/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8/(288000*2^(3/5)*Zeta(5)^(7/5)) - Zeta(3)/(12*2^(3/5)*Zeta(5)^(2/5))) * n^(2/5) + (Pi^4/(360*2^(2/5)*Zeta(5)^(3/5))) * n^(3/5) + 5*(Zeta(5)/2)^(1/5)/2 * n^(4/5)) * Zeta(5)^(173/1800) / (2^(26/225) * sqrt(5*Pi) * n^(1073/1800)). - Vaclav Kotesovec, Dec 08 2016

A279219 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(k+1)(2*k-1)/2).

Original entry on oeis.org

1, 1, 10, 40, 155, 560, 2051, 7080, 24064, 79370, 257067, 815593, 2545201, 7812699, 23639459, 70551216, 207932549, 605611061, 1744513262, 4973116444, 14038641287, 39263308551, 108849552289, 299248060986, 816159923366, 2209102273109, 5936069692320, 15840122529455, 41987363787469, 110584436073149

Offset: 0

Ilya Gutkovskiy, Dec 08 2016

nonn

Euler transform of the octagonal pyramidal numbers (A002414).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
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
Index to sequences related to pyramidal numbers

Cf. A000335, A002414, A279215, A279216, A279217, A279218.

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

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

a(n) ~ exp(-Zeta'(-1)/2 - Zeta(3)/(8*Pi^2) - Pi^16/(671846400000*Zeta(5)^3) - Pi^8*Zeta(3)/(5184000*Zeta(5)^2) - Zeta(3)^2/(240*Zeta(5)) + Zeta'(-3) + (Pi^12/(388800000*2^(3/5)*3^(1/5)*Zeta(5)^(11/5)) + Pi^4*Zeta(3)/(3600*2^(3/5) * 3^(1/5)*Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8/(432000*2^(1/5)*3^(2/5)*Zeta(5)^(7/5)) - Zeta(3)/(2^(11/5)*(3*Zeta(5))^(2/5))) * n^(2/5) + (Pi^4/(180*2^(4/5)*(3*Zeta(5))^(3/5))) * n^(3/5) + ((5*(3*Zeta(5))^(1/5))/(2^(7/5))) * n^(4/5)) * (3*Zeta(5))^(9/100) / (2^(23/100) * sqrt(5*Pi) * n^(59/100)). - Vaclav Kotesovec, Dec 08 2016

A317020 Expansion of Product_{k>=1} 1/(1 - x^k)^((5k-1)binomial(k+2,3)/4).

Original entry on oeis.org

1, 1, 10, 45, 185, 710, 2766, 10270, 37444, 132765, 462327, 1579563, 5311361, 17584084, 57414594, 185032557, 589183035, 1854974066, 5778722817, 17823440534, 54458411332, 164917654587, 495219323844, 1475145786950, 4360576440676, 12796007418881, 37287660835368, 107930276062786

Offset: 0

Ilya Gutkovskiy, Jul 19 2018

nonn

Euler transform of A002418.

N. J. A. Sloane, Transforms

Cf. A000391, A002418, A278769, A279218, A305653, A317017, A317019, A317021.

a:=series(mul(1/(1-x^k)^((5*k-1)*binomial(k+2,3)/4),k=1..100),x=0,28): seq(coeff(a,x,n),n=0..27); # Paolo P. Lava, Apr 02 2019

nmax = 27; CoefficientList[Series[Product[1/(1 - x^k)^((5 k - 1) Binomial[k + 2, 3]/4), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 27; CoefficientList[Series[Exp[Sum[x^k (1 + 4 x^k)/(k (1 - x^k)^5), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 (d + 1) (d + 2) (5 d - 1)/24, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 27}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A002418(k).
G.f.: exp(Sum_{k>=1} x^k*(1 + 4*x^k)/(k*(1 - x^k)^5)).
a(n) ~ (5/7)^(703/8640)/(2 * 3^(2143/2880) * n^(5023/8640) * Pi^(17/1440)) * exp(-1/144 + (1/12-Zeta'(-1))/12 - (21 * Zeta(3))/(400 * Pi^2) + (62921 * Zeta(5))/(80000 * Pi^4) - (194481 * Zeta(3) * Zeta(5)^2)/(50 * Pi^12) - (200120949 * Zeta(5)^3)/(1250 * Pi^14) + (28594081676916 * Zeta(5)^5)/(3125 * Pi^24) + (7 * Zeta'(-3))/12 + ((-343 * (7/5)^(1/6) * Pi)/(96000 * sqrt(3)) + (147 * (7/5)^(1/6) * sqrt(3) * Zeta(3) * Zeta(5))/(10 * Pi^7) + (1058841 * (7/5)^(1/6) * sqrt(3) * Zeta(5)^2)/(2000 * Pi^9) - (18211006359 * (7/5)^(1/6) * sqrt(3) * Zeta(5)^4)/(500 * Pi^19)) * n^(1/6) + (-((7/5)^(1/3) * Zeta(3))/(4 * Pi^2) - (1029 * (7/5)^(1/3) * Zeta(5))/(200 * Pi^4) + (10890936 * (7/5)^(1/3) * Zeta(5)^3)/(25 * Pi^14)) * n^(1/3) + ((7 * sqrt(7/15) * Pi)/120 - (9261 * sqrt(21/5) * Zeta(5)^2)/(5 * Pi^9)) * sqrt(n) + ((63 * (7/5)^(2/3) * Zeta(5))/(2 * Pi^4)) * n^(2/3) + ((2 * sqrt(3) * Pi)/(5^(5/6) * 7^(1/6))) * n^(5/6)). - Vaclav Kotesovec, Jul 28 2018

A318121 a(n) = [x^n] exp(Sum_{k>=1} x^k(1 + (n - 3)x^k)/(k*(1 - x^k)^4)).

Original entry on oeis.org

1, 1, 4, 15, 65, 269, 1205, 5325, 24064, 108849, 496790, 2275492, 10470720, 48325984, 223721404, 1038182441, 4828274432, 22497132116, 105001996350, 490816448220, 2297356108318, 10766317435860, 50511178395306, 237217429972191, 1115084064063866, 5246116796164594

Offset: 0

Ilya Gutkovskiy, Aug 18 2018

nonn

For n > 2, a(n) is the n-th term of the Euler transform of n-gonal pyramidal numbers.

N. J. A. Sloane, Transforms
Index to sequences related to pyramidal numbers

Cf. A000335, A279215, A279216, A279217, A279218, A279219, A318118.

Table[SeriesCoefficient[Exp[Sum[x^k (1 + (n - 3) x^k)/(k (1 - x^k)^4), {k, 1, n}]], {x, 0, n}], {n, 0, 25}]

a(n) ~ c * d^n / sqrt(n), where d = 4.80064986801984997726284... and c = 0.244706939300168165858... - Vaclav Kotesovec, Aug 19 2018

cp's OEIS Frontend

A279215 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(k+1)*(2*k+1)/6).

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A279216 Expansion of Product_{k>=1} 1/(1 - x^k)^(k^2*(k+1)/2).

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A279217 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(k+1)*(4*k-1)/6).

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A279219 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(k+1)*(2*k-1)/2).

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A317020 Expansion of Product_{k>=1} 1/(1 - x^k)^((5*k-1)*binomial(k+2,3)/4).

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A318121 a(n) = [x^n] exp(Sum_{k>=1} x^k*(1 + (n - 3)*x^k)/(k*(1 - x^k)^4)).

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A279215 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(k+1)(2*k+1)/6).

A279217 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(k+1)(4*k-1)/6).

A279219 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(k+1)(2*k-1)/2).

A317020 Expansion of Product_{k>=1} 1/(1 - x^k)^((5k-1)binomial(k+2,3)/4).

A318121 a(n) = [x^n] exp(Sum_{k>=1} x^k(1 + (n - 3)x^k)/(k*(1 - x^k)^4)).