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.

Showing 1-10 of 11 results. Next

A238016 Number A(n,k) of partitions of n^k into parts that are at most n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 5, 12, 5, 1, 1, 1, 9, 75, 64, 7, 1, 1, 1, 17, 588, 2280, 377, 11, 1, 1, 1, 33, 5043, 123464, 106852, 2432, 15, 1, 1, 1, 65, 44652, 7566280, 55567352, 6889527, 16475, 22, 1
Offset: 0

Views

Author

Alois P. Heinz, Feb 17 2014

Keywords

Comments

In general, for k>3, is column k asymptotic to exp(2*n) * n^((k-2)*n-k) / (2*Pi). For k=1 see A000041, for k=2 see A206226 and for k=3 see A238608. - Vaclav Kotesovec, May 25 2015
Conjecture: If f(n) >= O(n^4) then "number of partitions of f(n) into parts that are at most n" is asymptotic to f(n)^(n-1) / (n!*(n-1)!). See also A237998, A238000, A236810 or A258668-A258672. - Vaclav Kotesovec, Jun 07 2015

Examples

			A(3,1) = 3: 3, 21, 111.
A(3,2) = 12: 333, 3222, 3321, 22221, 32211, 33111, 222111, 321111, 2211111, 3111111, 21111111, 111111111.
A(2,3) = 5: 2222, 22211, 221111, 2111111, 11111111.
A(2,4) = 9: 22222222, 222222211, 2222221111, 22222111111, 222211111111, 2221111111111, 22111111111111, 211111111111111, 1111111111111111.
Square array A(n,k) begins:
  0, 1,   1,      1,        1,           1, ...
  1, 1,   1,      1,        1,           1, ...
  1, 2,   3,      5,        9,          17, ...
  1, 3,  12,     75,      588,        5043, ...
  1, 5,  64,   2280,   123464,     7566280, ...
  1, 7, 377, 106852, 55567352, 33432635477, ...

Links

Crossrefs

Columns k=0-10 give: A057427, A000041, A206226, A238608, A238609, A238610, A238611, A238612, A238613, A238614, A238615.
Rows n=0-10 give: A057427, A000012, A094373, A238630, A238631, A238632, A238633, A238634, A238635, A238636, A238637.
Main diagonal gives A238000.
Cf. A238010.

Programs

  • Mathematica
    A[n_, k_] := SeriesCoefficient[Product[1/(1-x^j), {j, 1, n}], {x, 0, n^k}]; A[0, 0] = 0; Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Oct 11 2015 *)

Formula

A(n,k) = [x^(n^k)] Product_{j=1..n} 1/(1-x^j).

A238000 Number of partitions of n^n into parts that are at most n.

Original entry on oeis.org

0, 1, 3, 75, 123464, 33432635477, 2561606354507677872, 85980297709044488588773397089, 1841159754991692001851990839259642586671980, 34687845413783594101366282545316028561007822069601179170488
Offset: 0

Views

Author

Alois P. Heinz, Feb 16 2014

Keywords

Examples

			a(1) = 1: 1.
a(2) = 3: 22, 211, 1111.
a(3) = 75: 333333333, ..., 111111111111111111111111111.

Links

Crossrefs

Main diagonal of A238010 and A238016.
Cf. A236810, A237512, A237998, A237999, A238001.

Programs

  • Mathematica
    a[n_] := SeriesCoefficient[Product[1/(1 - x^j), {j, 1, n}], {x, 0, n^n}];
a[0] = 0;
Table[a[n], {n, 0, 5}] (* Jean-François Alcover, Nov 03 2018 *)

Formula

a(n) = [x^(n^n)] Product_{j=1..n} 1/(1-x^j).
a(n) ~ exp(2*n) * n^(n*(n-3)) / (2*Pi). - Vaclav Kotesovec, May 25 2015

A237998 Number of partitions of 2^n into parts that are at most n.

Original entry on oeis.org

0, 1, 3, 10, 64, 831, 26207, 2239706, 567852809, 454241403975, 1192075219982204, 10510218491798860052, 315981966712495811700951, 32726459268483342710907384794, 11771239570056489326716955796095261, 14808470136486015545654676685321653888199
Offset: 0

Views

Author

Alois P. Heinz, Feb 16 2014

Keywords

Examples

			a(1) = 1: 11.
a(2) = 3: 22, 211, 1111.
a(3) = 10: 332, 2222, 3221, 3311, 22211, 32111, 221111, 311111, 2111111, 11111111.

Links

Crossrefs

Column k=2 of A238010.
Cf. A236810, A237512, A237999, A238000, A238001, A258672.

Programs

  • Mathematica
    a[n_] := SeriesCoefficient[Product[1/(1 - x^j), {j, 1, n}], {x, 0, 2^n}];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Nov 03 2018 *)

Formula

a(n) = [x^(2^n)] Product_{j=1..n} 1/(1-x^j).
a(n) ~ 2^(n*(n-1)) / (n!*(n-1)!). - Vaclav Kotesovec, Jun 05 2015

A237512 Number of solutions to Sum_{k=1..n} k*c(k) = n! , c(k) > 0.

Original entry on oeis.org

0, 1, 0, 1, 47, 55496, 2080571733, 4441900888487987, 849835826032526606030103, 20540228659655619974131131927286681, 82853643094578125257400348993596774353069331199, 70898139566455107685443806945119782661588205935442233026505921
Offset: 0

Views

Author

Wouter Meeussen, Feb 08 2014

Keywords

Comments

a(n) is the number of partitions of n! - n*(n+1)/2 into parts that are at most n. - Alois P. Heinz, Feb 08 2014

Links

Crossrefs

Cf. A236810.

Programs

  • Mathematica
    Table[Coefficient[Series[Product[x^k/(1-x^k),{k,n}],{x,0,n!}],x^(n!) ] ,{n,7}]

Formula

a(n) = [x^(n!)] Product_{k=1..n} x^k/(1-x^k).
a(n) = [x^(n!-n*(n+1)/2)] Product_{k=1..n} 1/(1-x^k). - Alois P. Heinz, Feb 08 2014
a(n) ~ n * (n!)^(n-3) ~ n^(n^2-5*n/2-1/2) * (2*Pi)^((n-3)/2) / exp(n*(n-3)-1/12). - Vaclav Kotesovec, Jun 05 2015

Extensions

a(8)-a(11) from Alois P. Heinz, Feb 08 2014

A237999 Number of partitions of 2^n into parts that are at most n with at least one part of each size.

Original entry on oeis.org

0, 1, 1, 2, 9, 119, 4935, 596763, 211517867, 224663223092, 734961197081208, 7614278809664610952, 256261752606028225485183, 28642174350851846128820426827, 10830277060032417592098008847162727, 14068379226083299071248895931891435683229
Offset: 0

Views

Author

Alois P. Heinz, Feb 16 2014

Keywords

Comments

From Gus Wiseman, May 31 2019: (Start)
Also the number of strict integer partitions of 2^n with n parts. For example, the a(1) = 1 through a(4) = 9 partitions are (A = 10):
(2) (31) (431) (6532)
(521) (6541)
(7432)
(7531)
(7621)
(8431)
(8521)
(9421)
(A321)
(End)

Examples

			a(1) = 1: 11.
a(2) = 1: 211.
a(3) = 2: 3221, 32111.
a(4) = 9: 433321, 443221, 4322221, 4332211, 4432111, 43222111, 43321111, 432211111, 4321111111.

Links

Crossrefs

Column k=2 of A238012.
Cf. A236810, A237512, A237998, A238000, A238001.
Cf. A000009, A002033, A067735, A126796, A283111.

Programs

  • Mathematica
    a[n_] := SeriesCoefficient[Product[1/(1 - x^j), {j, 1, n}], {x, 0, 2^n - n*(n + 1)/2}];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 15}] (* Jean-François Alcover, Aug 19 2018 *)

Formula

a(n) = [x^(2^n-n*(n+1)/2)] Product_{j=1..n} 1/(1-x^j).
a(n) ~ 2^(n*(n-1)) / (n!*(n-1)!). - Vaclav Kotesovec, Jun 05 2015

A258668 Number of partitions of (n-1)! into parts that are at most n.

Original entry on oeis.org

0, 1, 1, 2, 9, 333, 436140, 43079658188, 416768277321177570, 479533252041533318548951081, 82915824358567616469138110469031459201, 2733701379858734057121091262990514105871808751388190, 21440005073020480325389224271234447054906430340227337359088146738881
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 07 2015

Keywords

Links

Crossrefs

Cf. A236810, A237998, A238000, A258669, A258670, A258671.

Formula

a(n) ~ (n!)^(n-3) / n^(n-2).

A238001 Number of partitions of n^n into parts that are at most n with at least one part of each size.

Original entry on oeis.org

0, 1, 1, 48, 109809, 32796849930, 2555847904495965819, 85962759806610904434664386174, 1841132100297745277187328924904656111127054, 34687813181057391872792859998288408847592250236051615502024
Offset: 0

Views

Author

Alois P. Heinz, Feb 16 2014

Keywords

Examples

			a(1) = 1: 1.
a(2) = 1: 211.
a(3) = 48: 3333333321, ..., 321111111111111111111111.

Links

Crossrefs

Main diagonal of A238012.
Cf. A236810, A237512, A237998, A237999, A238000.

Programs

  • Mathematica
    maxExponent = 50; a[0] = 0; a[1] = 1;
a[n_] := Module[{}, aparts = List @@ (Product[1/(1 - x^j), {j, 1, n}] // Apart); cc = aparts + O[x]^maxExponent // CoefficientList[#, x]&; f[k_] = Total[FindSequenceFunction[#, k]& /@ cc]; f[n^n-n(n+1)/2 + 1] // Round];
Table[an = a[n]; Print[n, " ", an]; an, {n, 0, 9}] (* Jean-François Alcover, Nov 15 2018 *)

Formula

a(n) = [x^(n^n-n*(n+1)/2)] Product_{j=1..n} 1/(1-x^j).
a(n) ~ n^(n*(n-1)) / (n!*(n-1)!) ~ exp(2*n) * n^(n*(n-3)) / (2*Pi). - Vaclav Kotesovec, Jun 05 2015

A258669 Number of partitions of 2*n! into parts that are at most n.

Original entry on oeis.org

0, 1, 3, 19, 1033, 1302311, 74312057469, 291484874476601933, 109290159404495354765494065, 5262212497884462986538879797523944401, 42425405450182072688801993326226988336684453926401, 72600595215718916449806606426629386781199080157371905867835756161
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 07 2015

Keywords

Links

Crossrefs

Cf. A236810, A237998, A238000, A258668, A258670, A258671.

Formula

a(n) ~ n * 2^(n-1) * (n!)^(n-3).

A258670 Number of partitions of (2*n)! into parts that are at most n.

Original entry on oeis.org

0, 1, 13, 43561, 455366036161, 60209252317216962943201, 291857679749953126623181556402787323521, 120972618144269517756284629487432992029777542693069847287041
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 07 2015

Keywords

Comments

Conjecture: If f(n) >= O(n^4) then "number of partitions of f(n) into parts that are at most n" is asymptotic to f(n)^(n-1) / (n!*(n-1)!). For the examples see A238016 and A238010.

Links

Crossrefs

Cf. A236810, A237998, A238000, A238010, A238016, A258668, A258669, A258671.

Formula

a(n) ~ (2*n)!^(n-1) / (n!*(n-1)!).

A258671 Number of partitions of (n!)^2 into parts that are at most n.

Original entry on oeis.org

0, 1, 3, 127, 1361953, 14961046326601, 433366367372593816560481, 74029504174329565838647515081008812321, 147684970947386323832216294475743896349724799651361817601
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 07 2015

Keywords

Links

Crossrefs

Cf. A236810, A237998, A238000, A258668, A258669, A258670.

Formula

a(n) ~ n * (n!)^(2*n-4).
Showing 1-10 of 11 results. Next

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

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