A023871 -id:A023871 - cp's OEIS frontend

A285047 Expansion of Product_{k>=1} 1/(1-x^(k^2))^k.

1, 1, 1, 1, 3, 3, 3, 3, 6, 9, 9, 9, 13, 19, 19, 19, 28, 37, 43, 43, 57, 69, 81, 81, 100, 132, 150, 160, 184, 236, 260, 280, 319, 391, 460, 490, 565, 657, 771, 811, 922, 1084, 1243, 1363, 1510, 1781, 1985, 2185, 2388, 2775, 3159, 3439, 3832, 4335, 4963, 5323

Offset: 0

Vaclav Kotesovec, Apr 15 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000219, A001156, A023871, A266941, A281790.

nmax=100; CoefficientList[Series[Product[1/(1-x^(k^2))^k, {k,1,nmax}], {x,0,nmax}], x]

log(a(n)) ~ Pi*sqrt(n/3).

A363601 Number of partitions of n where there are k^2 - 1 kinds of parts k.

Original entry on oeis.org

1, 0, 3, 8, 21, 48, 126, 288, 693, 1568, 3570, 7896, 17417, 37632, 80823, 171192, 359733, 747936, 1543192, 3155760, 6407037, 12909024, 25835649, 51359136, 101470854, 199264128, 389096028, 755591256, 1459643343, 2805471984, 5366161740, 10216161336, 19362398580

Offset: 0

Seiichi Manyama, Jun 10 2023

Cf. A005380, A023871, A052847, A092348, A120844, A217093, A253289, A255271, A255802, A255803, A255835, A255836, A363602.

with(numtheory):
series(exp(add((sigma[3](k) - sigma[1](k))*x^k/k, k = 1..50)), x, 51):
seq(coeftayl(%, x = 0, n), n = 0..50); # Peter Bala, Jan 16 2025

my(N=40, x='x+O('x^N)); Vec(1/prod(k=1, N, (1-x^k)^(k^2-1)))

G.f.: 1/Product_{k>=1} (1-x^k)^(k^2-1).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} A092348(k) * a(n-k).
G.f.: exp(Sum_{k >= 1} (sigma_3(k) - sigma_1(k))*x^k/k) = 1 + 3*x^2 + 8*x^3 + 21*x^4 + 48*x^5 + .... - Peter Bala, Jan 16 2025

Name suggested by Joerg Arndt, Jun 11 2023

A363602 Number of partitions of n where there are k^2+1 kinds of parts k.

Original entry on oeis.org

1, 2, 8, 24, 72, 196, 532, 1368, 3467, 8520, 20580, 48664, 113330, 259588, 586692, 1308304, 2883427, 6283192, 13551344, 28940688, 61246052, 128492516, 267388008, 552126648, 1131750735, 2303690862, 4658080756, 9358912416, 18689701580, 37106245300, 73259451208

Offset: 0

Seiichi Manyama, Jun 10 2023

Cf. A023871, A092345, A363601.

my(N=40, x='x+O('x^N)); Vec(1/prod(k=1, N, (1-x^k)^(k^2+1)))

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

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} A092345(k) * a(n-k).

Name suggested by Joerg Arndt, Jun 11 2023

A285241 Expansion of Product_{k>=1} 1/(1 - k*x^k)^(k^2).

Original entry on oeis.org

1, 1, 9, 36, 140, 481, 1774, 5925, 20076, 64980, 208486, 652058, 2017023, 6117878, 18347256, 54222195, 158463794, 457570786, 1307951914, 3700153918, 10371860026, 28810051738, 79359812567, 216834266612, 587961817595, 1582612248239, 4230325722508

Offset: 0

Vaclav Kotesovec, Apr 15 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A006906, A023871, A266941, A285240, A285243.

nmax = 40; CoefficientList[Series[Product[1/(1-k*x^k)^(k^2), {k,1,nmax}], {x,0,nmax}], x]

a(n) ~ c * n^8 * 3^(n/3), where
if mod(n,3) = 0 then c = 3435237242728465092737309192093188152686332293\
03276380306112638865540880372901642880694943679256417087889777743957063\
209444405157397505005623042846150296486667845382334521513094023.8560142\
40331306860864399770618296475558098172993864629247911801570500913143642\
65158886200452165335605783203726486071335...
if mod(n,3) = 1 then c = 3435237242728465092737309192093188152686332293\
03276380306112638865540880372901642880694943679256417087889777743957063\
209444405157397505005623042846150296486667845382334521513094023.8560112\
77299895134841028015999951571187798033179513268954711586617617334007687\
07198348808962592621276659532114355538024...
if mod(n,3) = 2 then c = 3435237242728465092737309192093188152686332293\
03276380306112638865540880372901642880694943679256417087889777743957063\
209444405157397505005623042846150296486667845382334521513094023.8560117\
00278534968233203470801053870003971422069097966617636511346003845666735\
79293861331368526745743422198017148868212...
In closed form, a(n) ~ -(27*Product_{k>=4}((1 - k / 3^(k/3))^(-k^2)) / (13 + 128*3^(1/3) - 95*3^(2/3)) + 243*Product_{k>=4}((1 + (-1)^(1 + 2*k/3) * k / 3^(k/3))^(-k^2)) / ((-1)^(2*n/3) * ((3 + 2*(-3)^(1/3))^4 * (-3 + (-3)^(2/3)))) + (-1)^(1 - 4*n/3) * Product_{k>=4}((1 + (-1)^(1 + 4*k/3) * k / 3^(k/3))^(-k^2)) / ((1 + (-1/3)^(1/3)) * (1 - 2*(-1/3)^(2/3))^4)) / 793618560 * n^8 * 3^(n/3).

A300974 a(n) = [x^n] Product_{k>=1} 1/(1 - x^(k^2))^n.

Original entry on oeis.org

1, 1, 3, 10, 39, 151, 588, 2304, 9111, 36307, 145553, 586246, 2370264, 9614242, 39105580, 159444160, 651468967, 2666771488, 10934393619, 44899828056, 184616878289, 760010818689, 3132147583744, 12921037206764, 53351800567200, 220478125956426, 911839751015196, 3773836780169050

Offset: 0

Ilya Gutkovskiy, Mar 17 2018

nonn

Number of partitions of n into squares of n kinds.

Cf. A000290, A001156, A008485, A023871, A240944, A279225, A285047, A300975, A301518.

a:= proc(m) option remember; local b; b:= proc(n, i)
      option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(m+j-1, j)*b(n-i^2*j, i-1), j=0..n/i^2)))
      end: b(n, isqrt(n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 17 2018

Table[SeriesCoefficient[Product[1/(1 - x^k^2)^n, {k, 1, n}], {x, 0, n}], {n, 0, 27}]

From Vaclav Kotesovec, Mar 23 2018: (Start)
a(n) ~ c * d^n / sqrt(n), where
d = 4.216358447600641565890184638418336163396695730036... and
c = 0.26442245016754864773722176155288663999776... (End)

A305653 Expansion of Product_{k>=1} 1/(1 - x^k)^((k+1)*binomial(k+2,3)/2).

Original entry on oeis.org

1, 1, 7, 27, 98, 323, 1085, 3471, 10998, 33874, 102737, 305849, 897899, 2597822, 7423408, 20957775, 58524868, 161741013, 442705279, 1200718351, 3228796864, 8611973548, 22793714865, 59887897679, 156252738062, 404964879419, 1042884107691, 2669317020743, 6792321636929

Offset: 0

Ilya Gutkovskiy, Jun 07 2018

nonn

Euler transform of A002415, shifted left one place.

Alois P. Heinz, Table of n, a(n) for n = 0..2000
N. J. A. Sloane, Transforms

Cf. A000391, A002415, A023871, A279215, A290792.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(d^2*
      (d+2)*(d+1)^2/12, d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Jun 07 2018

nmax = 28; CoefficientList[Series[Product[1/(1 - x^k)^((k + 1) Binomial[k + 2, 3]/2), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 28; CoefficientList[Series[Exp[Sum[x^k (1 + 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)^2 (d + 2)/12, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 28}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A002415(k+1).
G.f.: exp(Sum_{k>=1} x^k*(1 + x^k)/(k*(1 - x^k)^5)).
a(n) ~ exp(Zeta'(-1)/6 - Zeta(3) / (4*Pi^2) + 149*Zeta(5) / (32*Pi^4) + 15876 * Zeta(3) * Zeta(5)^2 / Pi^12 - 666792 * Zeta(5)^3 / Pi^14 + 108884466432 * Zeta(5)^5 / (5*Pi^24) + Zeta'(-3)/3 + (-7*(7/2)^(1/6) * Pi / (384*sqrt(3)) - 21 * 2^(5/6) * sqrt(3) * 7^(1/6) * Zeta(3) * Zeta(5) / Pi^7 + 3087 * sqrt(3) * (7/2)^(1/6) * Zeta(5)^2 / (2*Pi^9) - 30339036 * 2^(5/6) * sqrt(3) * 7^(1/6) * Zeta(5)^4 / Pi^19) * n^(1/6) + ((7/2)^(1/3) * Zeta(3) / (2*Pi^2) - 21 * (7/2)^(1/3) * Zeta(5) / (2*Pi^4) + 254016 * 2^(2/3) * 7^(1/3) * Zeta(5)^3 / Pi^14) * n^(1/3) + (sqrt(7/6) * Pi / 12 - 756 * sqrt(42) * Zeta(5)^2 / Pi^9) * sqrt(n) + (9 * 2^(1/3) * 7^(2/3) * Zeta(5) / Pi^4) * n^(2/3) + (2 * (2/7)^(1/6) * sqrt(3) * Pi) / 5 * n^(5/6)) * Pi^(1/90) / (2^(247/270) * 3^(34/45) * 7^(23/270) * n^(79/135)). - Vaclav Kotesovec, Jun 08 2018

A274998 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(3k-2)).

Original entry on oeis.org

1, 1, 9, 30, 106, 339, 1106, 3355, 10102, 29358, 83908, 234394, 644286, 1739933, 4631675, 12153197, 31485413, 80576160, 203902261, 510490213, 1265353568, 3106771717, 7559844833, 18239351931, 43650061720, 103657177941, 244346681972, 571930478187, 1329655624297, 3071230379625, 7049750442386, 16085170634548, 36489192684910

Offset: 0

Ilya Gutkovskiy, Nov 30 2016

Euler transform of the octagonal numbers (A000567).

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002.
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, Octagonal Number
Index to sequences related to polygonal numbers

Cf. A000294, A000567, A000335, A023871, A278768, A294667, A294691, A294692.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d^2*(3*d-2), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 02 2016

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

from sympy import divisors
from sympy.core.cache import cacheit
@cacheit
def a(n): return 1 if n==0 else sum(sum(d**2*(3*d - 2) for d in divisors(j))*a(n - j) for j in range(1, n + 1))//n
print([a(n) for n in range(51)]) # Indranil Ghosh, Aug 06 2017, after Maple code

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

a(n) ~ exp(4*Pi*n^(3/4) / (3*5^(1/4)) - 2*Zeta(3) * sqrt(5*n) / Pi^2 - 10*Zeta(3)^2 * (5*n)^(1/4) / Pi^5 - 200*Zeta(3)^3 / (3*Pi^8) - 3*Zeta(3) / (4*Pi^2) - 1/6) * A^2 / (2^(3/2) * 5^(1/12) * Pi^(1/6) * n^(7/12)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 08 2017

A294750 Expansion of Product_{k>=1} 1/(1 - x^(2*k - 1))^(k^2).

Original entry on oeis.org

1, 1, 1, 5, 5, 14, 24, 40, 76, 121, 230, 356, 635, 1024, 1709, 2820, 4510, 7430, 11712, 19007, 29800, 47490, 74261, 116385, 181423, 280696, 434956, 666970, 1025816, 1562504, 2383916, 3611493, 5467505, 8241296, 12389888, 18581326, 27765501, 41426994, 61573390

Offset: 0

Vaclav Kotesovec, Nov 08 2017

nonn

In general, if g.f. = Product_{k>=1} 1/(1 - x^(2*k-1))^(c2*k^2 + c1*k + c0) and c2>0, then a(n) ~ exp(2*Pi/3 * (2*c2/15)^(1/4) * n^(3/4) + (c1+c2) * Zeta(3) / Pi^2 * sqrt(15*n/(2*c2)) + (Pi*(4*c0 + 2*c1 + c2)/24 - 15*(c1+c2)^2 * Zeta(3)^2 / (2*c2*Pi^5)) * (15*n/(2*c2))^(1/4) + 75*(c1+c2)^3 * Zeta(3)^3 / (c2^2 * Pi^8) - (5*c0 + 15*c1/4 + c2/2 + 5*c1*(2*c0 + c1) / (2*c2)) * (Zeta(3) / (4*Pi^2)) - (c1+c2)/24) * A^((c1+c2)/2) * (15/c2)^((c1+c2)/96 - 1/8) * n^((c1+c2)/96 - 5/8) / (2^(15/8 + c0/2 + (29*c1 + 17*c2)/96) * Pi^((c1+c2)/24)), where A is the Glaisher-Kinkelin constant A074962.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A023871, A035528, A294749, A294755.

nmax = 50; CoefficientList[Series[Product[1/(1-x^(2*k-1))^(k^2), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(2*Pi/3 * (2/15)^(1/4) * n^(3/4) + Zeta(3) * sqrt(15*n/2) / Pi^2 + (Pi * (15/2)^(1/4)/24 - Zeta(3)^2 * (15/2)^(5/4) / Pi^5) * n^(1/4) + 75*Zeta(3)^3 / Pi^8 - Zeta(3) / (8*Pi^2) - 1/24) * sqrt(A) / (2^(197/96) * 15^(11/96) * Pi^(1/24) * n^(59/96)), where A is the Glaisher-Kinkelin constant A074962.

A318118 a(n) = [x^n] exp(Sum_{k>=1} x^k(1 + (n - 3)x^k)/(k*(1 - x^k)^3)).

Original entry on oeis.org

1, 1, 3, 10, 40, 150, 616, 2456, 10102, 41400, 171526, 712111, 2972115, 12434993, 52195414, 219567909, 925704792, 3909841659, 16541598215, 70085877919, 297347922785, 1263046810334, 5370930049915, 22861883482838, 97402827429118, 415332438952517, 1772380322197432

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 numbers.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms
Index to sequences related to polygonal numbers

Cf. A000294, A023871, A274998, A278767, A278768, A278769.

Table[SeriesCoefficient[Exp[Sum[x^k (1 + (n - 3) x^k)/(k (1 - x^k)^3), {k, 1, n}]], {x, 0, n}], {n, 0, 26}]

a(n) ~ c * d^n / sqrt(n), where d = 4.3505530790182509701639869563721679988879373943131559534408716195123... and c = 0.2276354216252041005336767937139336687746108521151301186102034... - Vaclav Kotesovec, Aug 18 2018

A328407 G.f. A(x) satisfies: A(x) = A(x^2) + x * (1 + x) / (1 - x)^3.

Original entry on oeis.org

1, 5, 9, 21, 25, 45, 49, 85, 81, 125, 121, 189, 169, 245, 225, 341, 289, 405, 361, 525, 441, 605, 529, 765, 625, 845, 729, 1029, 841, 1125, 961, 1365, 1089, 1445, 1225, 1701, 1369, 1805, 1521, 2125, 1681, 2205, 1849, 2541, 2025, 2645, 2209, 3069, 2401, 3125, 2601, 3549, 2809, 3645, 3025

Offset: 1

Ilya Gutkovskiy, Oct 14 2019

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000290, A001511, A007434, A016754, A023871, A129527, A209229, A328408.

[n eq 1 select 1 else IsOdd(n) select n^2 else Self(n div 2)+n^2 :n in [1..55]]; // Marius A. Burtea, Oct 15 2019

nmax = 55; CoefficientList[Series[Sum[x^(2^k) (1 + x^(2^k))/(1 - x^(2^k))^3, {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x] // Rest
a[n_] := If[EvenQ[n], a[n/2] + n^2, n^2]; Table[a[n], {n, 1, 55}]
Table[DivisorSum[n, Boole[IntegerQ[Log[2, n/#]]] #^2 &], {n, 1, 55}]
f[p_, e_] := If[p == 2, (4^(e + 1) - 1)/3, p^(2*e)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *)

G.f.: Sum_{k>=0} x^(2^k) * (1 + x^(2^k)) / (1 - x^(2^k))^3.
G.f.: (1/3) * Sum_{k>=1} J_2(2*k) * x^k / (1 - x^k), where J_2() is the Jordan function (A007434).
Dirichlet g.f.: zeta(s-2) / (1 - 2^(-s)).
a(2*n) = a(n) + 4*n^2, a(2*n+1) = (2*n + 1)^2.
a(n) = Sum_{d|n} A209229(n/d) * d^2.
Product_{n>=1} (1 + x^n)^a(n) = g.f. for A023871.
Sum_{k=1..n} a(k) ~ 8*n^3/21. - Vaclav Kotesovec, Oct 15 2019
Multiplicative with a(2^e) = (4^(e+1)-1)/3, and a(p^e) = p^(2*e) for an odd prime p. - Amiram Eldar, Oct 25 2020

cp's OEIS Frontend

A285047 Expansion of Product_{k>=1} 1/(1-x^(k^2))^k.

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A363601 Number of partitions of n where there are k^2 - 1 kinds of parts k.

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A363602 Number of partitions of n where there are k^2+1 kinds of parts k.

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A285241 Expansion of Product_{k>=1} 1/(1 - k*x^k)^(k^2).

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A300974 a(n) = [x^n] Product_{k>=1} 1/(1 - x^(k^2))^n.

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A305653 Expansion of Product_{k>=1} 1/(1 - x^k)^((k+1)*binomial(k+2,3)/2).

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A274998 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(3*k-2)).

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A294750 Expansion of Product_{k>=1} 1/(1 - x^(2*k - 1))^(k^2).

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A274998 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(3k-2)).

A318118 a(n) = [x^n] exp(Sum_{k>=1} x^k(1 + (n - 3)x^k)/(k*(1 - x^k)^3)).