cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A022731 Expansion of Product_{m>=1} 1/(1 - m*q^m)^7.

Original entry on oeis.org

1, 7, 42, 203, 889, 3535, 13209, 46551, 156905, 507787, 1588594, 4819003, 14231294, 41007134, 115589904, 319284693, 865781826, 2307766118, 6054769679, 15652436765, 39909873983, 100451866962
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=7 of A297328.

Programs

  • Magma
    n:=50; R:=PowerSeriesRing(Integers(), n); Coefficients(R!(&*[(1/(1-m*x^m))^7:m in [1..n]])); // G. C. Greubel, Jul 25 2018
  • Mathematica
    With[{nmax = 50}, CoefficientList[Series[Product[(1 - k*q^k)^-7, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Jul 25 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1-n*q^n)^-7)) \\ G. C. Greubel, Jul 25 2018

Formula

G.f.: exp(7*Sum_{j>=1} Sum_{k>=1} k^j*x^(j*k)/j). - Ilya Gutkovskiy, Feb 07 2018

A022732 Expansion of Product_{m>=1} 1/(1 - m*q^m)^8.

Original entry on oeis.org

1, 8, 52, 272, 1274, 5408, 21448, 80080, 285043, 972496, 3200644, 10199456, 31592350, 95366176, 281269560, 812094448, 2299480441, 6394796832, 17489643664, 47096042032, 124993380566, 327249781952
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=8 of A297328.

Programs

  • Magma
    n:=50; R:=PowerSeriesRing(Integers(), n); Coefficients(R!(&*[(1/(1-m*x^m))^8:m in [1..n]])); // G. C. Greubel, Jul 25 2018
  • Mathematica
    With[{nmax = 50}, CoefficientList[Series[Product[(1 - k*q^k)^-8, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Jul 25 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1-n*q^n)^-8)) \\ G. C. Greubel, Jul 25 2018

Formula

G.f.: exp(8*Sum_{j>=1} Sum_{k>=1} k^j*x^(j*k)/j). - Ilya Gutkovskiy, Feb 07 2018

A022733 Expansion of Product_{m>=1} 1/(1 - m*q^m)^9.

Original entry on oeis.org

1, 9, 63, 354, 1764, 7947, 33294, 131049, 490437, 1756243, 6055749, 20190402, 65342031, 205853535, 632948256, 1903369146, 5608257129, 16216492509, 46080035361, 128829484620, 354757096107, 963099596421
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Column k=9 of A297328.

Programs

  • Magma
    n:=50; R:=PowerSeriesRing(Integers(), n); Coefficients(R!(&*[(1/(1-m*x^m))^9:m in [1..n]])); // G. C. Greubel, Jul 25 2018
  • Mathematica
    With[{nmax = 50}, CoefficientList[Series[Product[(1 - k*q^k)^-9, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Jul 25 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1-n*q^n)^-9)) \\ G. C. Greubel, Jul 25 2018

Formula

G.f.: exp(9*Sum_{j>=1} Sum_{k>=1} k^j*x^(j*k)/j). - Ilya Gutkovskiy, Feb 07 2018

A022736 Expansion of 1/Product_{m>=1} (1 - m*q^m)^12.

Original entry on oeis.org

1, 12, 102, 688, 4029, 21156, 102246, 461448, 1967658, 7990996, 31110432, 116685288, 423366831, 1490904528, 5110173678, 17088259888, 55862240688, 178836472032, 561532752086, 1731639278904, 5250722230962
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 12, g(n) = n. - Seiichi Manyama, Aug 16 2023

Links

Crossrefs

Column k=12 of A297328.
Cf. A078308.

Programs

  • Magma
    n:=50; R:=PowerSeriesRing(Integers(), n); Coefficients(R!(&*[(1/(1-m*x^m))^12:m in [1..n]])); // G. C. Greubel, Jul 25 2018
  • Mathematica
    With[{nmax = 50}, CoefficientList[Series[Product[(1 - k*q^k)^-12, {k, 1, nmax}], {q, 0, nmax}], q]] (* G. C. Greubel, Jul 25 2018 *)
  • PARI
    m=50; q='q+O('q^m); Vec(prod(n=1,m,(1-n*q^n)^-12)) \\ G. C. Greubel, Jul 25 2018

Formula

a(0) = 1; a(n) = (12/n) * Sum_{k=1..n} A078308(k) * a(n-k). - Seiichi Manyama, Aug 16 2023

A022739 Expansion of Product (1-m*q^m)^-15; m=1..infinity.

Original entry on oeis.org

1, 15, 150, 1175, 7875, 46953, 255745, 1293825, 6154530, 27772930, 119720802, 495673410, 1979777985, 7656005115, 28752006375, 105129718102, 375082371420, 1308260532180, 4468338006465
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A006906, A022610, A022643, A023013.
Column k=15 of A297328.

Programs

  • Maple
    N:= 30: # for a(0)..a(N)
P:= mul(1-m*q^m,m=1..N):
S:= series(P,q,N+1):
S15:= series(S^(-15),q,N+1):
seq(coeff(S15,q,i),i=0..N); # Robert Israel, Dec 22 2019
  • Mathematica
    nmax = 20; CoefficientList[Series[Product[1/(1 - k*x^k)^15, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 23 2019 *)

A022740 Expansion of Product (1-m*q^m)^-16; m=1..inf.

Original entry on oeis.org

1, 16, 168, 1376, 9604, 59488, 335904, 1758816, 8646986, 40281296, 179065184, 763837600, 3140732344, 12494160288, 48236274976, 181203877248, 663837626163, 2376282980272, 8325497904672
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A006906, A022611, A022644, A023014.
Column k=16 of A297328.

Programs

  • Maple
    N:= 30: # for a(0)..a(N)
P:= mul(1-m*q^m,m=1..N):
S:= series(P,q,N+1):
S16:= series(S^(-16),q,N+1):
seq(coeff(S16,q,i),i=0..N); # Robert Israel, Dec 22 2019
  • Mathematica
    nmax = 20; CoefficientList[Series[Product[1/(1 - k*x^k)^16, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 23 2019 *)

A022741 Expansion of 1/Product_{m>=1} (1 - m*q^m)^17.

Original entry on oeis.org

1, 17, 187, 1598, 11594, 74477, 435268, 2355265, 11950456, 57389960, 262741086, 1153256370, 4875584955, 19928324460, 79000508546, 304554050669, 1144374191351, 4199521413019, 15076785650771
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 17, g(n) = n. - Seiichi Manyama, Aug 16 2023

Links

Crossrefs

Column k=17 of A297328.
Cf. A078308.

Formula

a(0) = 1; a(n) = (17/n) * Sum_{k=1..n} A078308(k) * a(n-k). - Seiichi Manyama, Aug 16 2023

A022742 Expansion of 1/Product_{m>=1} (1 - m*q^m)^18.

Original entry on oeis.org

1, 18, 207, 1842, 13869, 92250, 557214, 3111624, 16272972, 80461694, 378917667, 1709416008, 7422200694, 31136646366, 126608628735, 500368588830, 1926482319873, 7240422289338, 26610335585263
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 18, g(n) = n. - Seiichi Manyama, Aug 16 2023

Links

Crossrefs

Column k=18 of A297328.
Cf. A078308.

Formula

a(0) = 1; a(n) = (18/n) * Sum_{k=1..n} A078308(k) * a(n-k). - Seiichi Manyama, Aug 16 2023

A022743 Expansion of 1/Product_{m>=1} (1 - m*q^m)^19.

Original entry on oeis.org

1, 19, 228, 2109, 16454, 113164, 705527, 4060870, 21863490, 111178196, 537978635, 2491812228, 11100464810, 47746888432, 198952868210, 805306011651, 3174012174213, 12206318615473, 45884592463642
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 19, g(n) = n. - Seiichi Manyama, Aug 16 2023

Links

Crossrefs

Column k=19 of A297328.
Cf. A078308.

Formula

a(0) = 1; a(n) = (19/n) * Sum_{k=1..n} A078308(k) * a(n-k). - Seiichi Manyama, Aug 16 2023

A022744 Expansion of 1/Product_{m>=1} (1 - m*q^m)^20.

Original entry on oeis.org

1, 20, 250, 2400, 19375, 137604, 884430, 5241000, 29017815, 151597440, 752997538, 3577442200, 16335561280, 71979549460, 307075300540, 1271938667704, 5127690095315, 20161295885120, 77454572685460
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 20, g(n) = n. - Seiichi Manyama, Aug 16 2023

Links

Crossrefs

Column k=20 of A297328.
Cf. A078308.

Formula

a(0) = 1; a(n) = (20/n) * Sum_{k=1..n} A078308(k) * a(n-k). - Seiichi Manyama, Aug 16 2023
Previous Showing 11-20 of 32 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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