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.

Previous Showing 11-19 of 19 results.

A014161 Apply partial sum operator 4 times to partition numbers.

Original entry on oeis.org

1, 5, 16, 41, 92, 188, 359, 650, 1128, 1890, 3075, 4878, 7571, 11527, 17254, 25436, 36988, 53122, 75435, 106014, 147573, 203618, 278657, 378453, 510344, 683626, 910031, 1204301, 1584896, 2074841, 2702765
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

A014161 convolved with A010815 = the tetrahedral numbers: 1, 4, 10, 20, 35, ... . - Gary W. Adamson, Nov 09 2008

Links

Crossrefs

Cf. A000041.
Cf. A010815. - Gary W. Adamson, Nov 09 2008
Column k=5 of A292508.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[1/((1-x)^4 * Product[1-x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 30 2015 *)

Formula

a(n) ~ 3^(3/2)*n * exp(Pi*sqrt(2*n/3)) / Pi^4. - Vaclav Kotesovec, Oct 30 2015
G.f.: 1/(1-x)^4 * Product_{k>=1} 1/(1-x^k). - Vaclav Kotesovec, Oct 30 2015

A120477 Apply partial sum operator 5 times to partition numbers.

Original entry on oeis.org

1, 6, 22, 63, 155, 343, 702, 1352, 2480, 4370, 7445, 12323, 19894, 31421, 48675, 74111, 111099, 164221, 239656, 345670, 493243, 696861, 975518, 1353971, 1864315, 2547941, 3457972, 4662273, 6247169, 8322010, 11024775, 14528914, 19051697
Offset: 0

Views

Author

Jonathan Vos Post, Jul 21 2006

Keywords

Comments

In general, if g.f. = 1/(1-x)^m * Product_{k>=1} 1/(1-x^k), then a(n) ~ 2^(m/2 - 2) * 3^((m-1)/2) * n^(m/2 - 1) * exp(Pi*sqrt(2*n/3)) / Pi^m. - Vaclav Kotesovec, Oct 30 2015

Links

Crossrefs

Cf. A000041, A000070, A014153, A014160, A014161.
Column k=6 of A292508.

Programs

  • Maple
    with(combinat): g:=1/(1-x)^5/product(1-x^k,k=1..50): gser:=series(g,x=0,40): seq(coeff(gser,x,n),n=0..37); # Emeric Deutsch, Jul 24 2006
  • Mathematica
    nmax = 50; CoefficientList[Series[1/((1-x)^5 * Product[1-x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 30 2015 *)

Formula

G.f.: 1/((1-x)^5*Product_{k>=1} (1-x^k)). - Emeric Deutsch, Jul 24 2006
a(n) ~ 9*sqrt(2)*n^(3/2) * exp(Pi*sqrt(2*n/3)) / Pi^5. - Vaclav Kotesovec, Oct 30 2015

Extensions

More terms from Emeric Deutsch, Jul 24 2006

A346424 Number of partitions of the 2n-multiset {0,...,0,1,2,...,n}.

Original entry on oeis.org

1, 2, 11, 74, 592, 5317, 52902, 572402, 6670707, 83025806, 1096662664, 15292076689, 224145880470, 3440981816071, 55153081768896, 920494136057715, 15959177281931953, 286834809549486462, 5334308665713522860, 102476857445135062727, 2030589375575413246579
Offset: 0

Views

Author

Alois P. Heinz, Jul 16 2021

Keywords

Comments

Also number of factorizations of 2^n * Product_{i=1..n} prime(i+1); a(2) = 11: 2*2*3*5, 3*4*5, 2*5*6, 6*10, 2*3*10, 5*12, 4*15, 2*2*15, 3*20, 2*30, 60.

Examples

			a(0) = 1: {}.
a(1) = 2: 01, 0|1.
a(2) = 11: 00|1|2, 001|2, 1|002, 0|0|1|2, 0|01|2, 0|1|02, 01|02, 00|12, 0|0|12, 0|012, 0012.

Links

Crossrefs

Main diagonal of A346426.
Cf. A000040, A000041, A000079, A000110, A001055, A048993, A070826, A292508, A346519.

Programs

  • Maple
    s:= proc(n) option remember; expand(`if`(n=0, 1,
      x*add(s(n-j)*binomial(n-1, j-1), j=1..n)))
    end:
S:= proc(n, k) option remember; coeff(s(n), x, k) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=0,
      combinat[numbpart](n), add(b(n-j, i-1), j=0..n)))
    end:
a:= n-> add(S(n, j)*b(n, j), j=0..n):
seq(a(n), n=0..21);
  • Mathematica
    s[n_] := s[n] = Expand[If[n == 0, 1,
     x*Sum[s[n - j]*Binomial[n - 1, j - 1], {j, 1, n}]]];
S[n_, k_] := S[n, k] = Coefficient[s[n], x, k];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 0,
     PartitionsP[n], Sum[b[n - j, i - 1], {j, 0, n}]]];
a[n_] := Sum[S[n, j]*b[n, j], {j, 0, n}];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Apr 06 2022, after Alois P. Heinz *)

Formula

a(n) = A001055(A000079(n)*A070826(n+1)).
a(n) = Sum_{j=0..n} A048993(n,j)*A292508(n,j+1).
a(n) = A346426(n,n).

A320753 Number of partitions of n with seven kinds of 1.

Original entry on oeis.org

1, 7, 29, 92, 247, 590, 1292, 2644, 5124, 9494, 16939, 29262, 49156, 80577, 129252, 203363, 314462, 478683, 718339, 1064009, 1557252, 2254113, 3229631, 4583602, 6447917, 8995858, 12453830, 17116103, 23363272, 31685282, 42710057, 57238971, 76290668, 101155025
Offset: 0

Views

Author

Alois P. Heinz, Oct 20 2018

Keywords

Links

Crossrefs

Column k=7 of A292508.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-x)^7*(&*[1-x^j: j in [2..30]])))); // G. C. Greubel, Oct 27 2018
  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      (numtheory[sigma](j)+6)*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);
  • Mathematica
    nmax = 50; CoefficientList[Series[1/((1-x)^6 * Product[1-x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 24 2018 *)
  • PARI
    x='x+O('x^30); Vec(1/((1-x)^7*prod(j=2, 40, 1-x^j))) \\ G. C. Greubel, Oct 27 2018

Formula

G.f.: 1/(1-x)^7 * 1/Product_{j>1} (1-x^j).
Euler transform of 7,1,1,1,... .
a(n) ~ 2 * 3^(5/2) * n^2 * exp(Pi*sqrt(2*n/3)) / Pi^6. - Vaclav Kotesovec, Oct 24 2018

A320754 Number of partitions of n with eight kinds of 1.

Original entry on oeis.org

1, 8, 37, 129, 376, 966, 2258, 4902, 10026, 19520, 36459, 65721, 114877, 195454, 324706, 528069, 842531, 1321214, 2039553, 3103562, 4660814, 6914927, 10144558, 14728160, 21176077, 30171935, 42625765, 59741868, 83105140, 114790422, 157500479, 214739450
Offset: 0

Views

Author

Alois P. Heinz, Oct 20 2018

Keywords

Links

Crossrefs

Column k=8 of A292508.

Programs

  • Magma
    m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-x)^8*(&*[1-x^j: j in [2..30]])))); // G. C. Greubel, Oct 27 2018
  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      (numtheory[sigma](j)+7)*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);
  • Mathematica
    nmax = 50; CoefficientList[Series[1/((1-x)^7 * Product[1-x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 24 2018 *)
  • PARI
    x='x+O('x^40); Vec(1/((1-x)^8*prod(j=2, 40, 1-x^j))) \\ G. C. Greubel, Oct 27 2018

Formula

G.f.: 1/(1-x)^8 * 1/Product_{j>1} (1-x^j).
Euler transform of 8,1,1,1,... .
a(n) ~ 2^(3/2) * 3^3 * n^(5/2) * exp(Pi*sqrt(2*n/3)) / Pi^7. - Vaclav Kotesovec, Oct 24 2018

A320755 Number of partitions of n with nine kinds of 1.

Original entry on oeis.org

1, 9, 46, 175, 551, 1517, 3775, 8677, 18703, 38223, 74682, 140403, 255280, 450734, 775440, 1303509, 2146040, 3467254, 5506807, 8610369, 13271183, 20186110, 30330668, 45058828, 66234905, 96406840, 139032605, 198774473, 281879613, 396670035, 554170514, 768909964
Offset: 0

Views

Author

Alois P. Heinz, Oct 20 2018

Keywords

Links

Crossrefs

Column k=9 of A292508.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-x)^9*(&*[1-x^j: j in [2..30]])))); // G. C. Greubel, Oct 27 2018
  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      (numtheory[sigma](j)+8)*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);
  • Mathematica
    nmax = 50; CoefficientList[Series[1/((1-x)^8 * Product[1-x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 24 2018 *)
  • PARI
    x='x+O('x^30); Vec(1/((1-x)^9*prod(j=2, 40, 1-x^j))) \\ G. C. Greubel, Oct 27 2018

Formula

G.f.: 1/(1-x)^9 * 1/Product_{j>1} (1-x^j).
Euler transform of 9,1,1,1,... .
a(n) ~ 4 * 3^(7/2) * n^3 * exp(Pi*sqrt(2*n/3)) / Pi^8. - Vaclav Kotesovec, Oct 24 2018

A320756 Number of partitions of n with ten kinds of 1.

Original entry on oeis.org

1, 10, 56, 231, 782, 2299, 6074, 14751, 33454, 71677, 146359, 286762, 542042, 992776, 1768216, 3071725, 5217765, 8685019, 14191826, 22802195, 36073378, 56259488, 86590156, 131648984, 197883889, 294290729, 433323334, 632097807, 913977420, 1310647455, 1864817969
Offset: 0

Views

Author

Alois P. Heinz, Oct 20 2018

Keywords

Links

Crossrefs

Column k=10 of A292508.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-x)^10*(&*[1-x^j: j in [2..30]])))); // G. C. Greubel, Oct 27 2018
  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      (numtheory[sigma](j)+9)*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);
  • Mathematica
    nmax = 50; CoefficientList[Series[1/((1-x)^9 * Product[1-x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 24 2018 *)
  • PARI
    x='x+O('x^30); Vec(1/((1-x)^10*prod(j=2, 40, 1-x^j))) \\ G. C. Greubel, Oct 27 2018

Formula

G.f.: 1/(1-x)^10 * 1/Product_{j>1} (1-x^j).
Euler transform of 10,1,1,1,... .
a(n) ~ 2^(5/2) * 3^4 * n^(7/2) * exp(Pi*sqrt(2*n/3)) / Pi^9. - Vaclav Kotesovec, Oct 24 2018

A325951 G.f.: 1/(1-x)^3 * Product_{k>=1} (1 + x^k).

Original entry on oeis.org

1, 4, 10, 21, 39, 67, 109, 170, 256, 375, 537, 754, 1041, 1416, 1901, 2523, 3314, 4312, 5563, 7121, 9050, 11426, 14338, 17890, 22204, 27422, 33709, 41257, 50288, 61058, 73863, 89043, 106988, 128146, 153029, 182222, 216393, 256302, 302813, 356908, 419700
Offset: 0

Views

Author

Vaclav Kotesovec, May 28 2019

Keywords

Crossrefs

Cf. A000009, A014160, A036469, A095944, A120477, A292508.

Programs

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

Formula

a(n) ~ 2 * 3^(5/4) * n^(3/4) * exp(Pi*sqrt(n/3)) / Pi^3.

A325952 G.f.: 1/(1-x)^4 * Product_{k>=1} (1 + x^k).

Original entry on oeis.org

1, 5, 15, 36, 75, 142, 251, 421, 677, 1052, 1589, 2343, 3384, 4800, 6701, 9224, 12538, 16850, 22413, 29534, 38584, 50010, 64348, 82238, 104442, 131864, 165573, 206830, 257118, 318176, 392039, 481082, 588070, 716216, 869245, 1051467, 1267860, 1524162, 1826975
Offset: 0

Views

Author

Vaclav Kotesovec, May 28 2019

Keywords

Comments

In general, if g.f. = 1/(1-x)^m * Product_{k>=1} (1 + x^k), then a(n) ~ 2^(m - 2) * 3^(m/2 - 1/4) * n^(m/2 - 3/4) * exp(Pi*sqrt(n/3)) / Pi^m.

Crossrefs

Cf. A000009, A014161, A036469, A095944, A120477, A292508, A325951.

Programs

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

Formula

a(n) ~ 4 * 3^(7/4) * n^(5/4) * exp(Pi*sqrt(n/3)) / Pi^4.
Previous Showing 11-19 of 19 results.

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

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