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 12 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).

A304176 Number of partitions of n^3 into exactly n parts.

Original entry on oeis.org

1, 1, 4, 61, 1906, 91606, 6023602, 505853354, 51900711796, 6306147384659, 886745696653253, 141778041323736643, 25417656781153090889, 5052180112449982704619, 1103058286595668300801794, 262487324530101028337614478, 67628783852463631751658038290
Offset: 0

Views

Author

Seiichi Manyama, May 07 2018

Keywords

Examples

			n | Partitions of n^3 into exactly n parts
--+------------------------------------------------------------
1 |   1.
2 |   7+1 = 6+2 = 5+3 = 4+4.
3 |   25+ 1+1 = 24+ 2+1 = 23+ 3+1 = 23+ 2+2 = 22+ 4+1 = 22+ 3+2
  | = 21+ 5+1 = 21+ 4+2 = 21+ 3+3 = 20+ 6+1 = 20+ 5+2 = 20+ 4+3
  | = 19+ 7+1 = 19+ 6+2 = 19+ 5+3 = 19+ 4+4 = 18+ 8+1 = 18+ 7+2
  | = 18+ 6+3 = 18+ 5+4 = 17+ 9+1 = 17+ 8+2 = 17+ 7+3 = 17+ 6+4
  | = 17+ 5+5 = 16+10+1 = 16+ 9+2 = 16+ 8+3 = 16+ 7+4 = 16+ 6+5
  | = 15+11+1 = 15+10+2 = 15+ 9+3 = 15+ 8+4 = 15+ 7+5 = 15+ 6+6
  | = 14+12+1 = 14+11+2 = 14+10+3 = 14+ 9+4 = 14+ 8+5 = 14+ 7+6
  | = 13+13+1 = 13+12+2 = 13+11+3 = 13+10+4 = 13+ 9+5 = 13+ 8+6
  | = 13+ 7+7 = 12+12+3 = 12+11+4 = 12+10+5 = 12+ 9+6 = 12+ 8+7
  | = 11+11+5 = 11+10+6 = 11+ 9+7 = 11+ 8+8 = 10+10+7 = 10+ 9+8
  | =  9+ 9+9.

Links

Crossrefs

Cf. A206240, A238608, A304208, A304212.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1)+b(n-i, min(i, n-i)))
    end:
a:= n-> b(n^3-n, n):
seq(a(n), n=0..20);  # Alois P. Heinz, May 07 2018
  • Mathematica
    $RecursionLimit = 2000;
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, b[n, i - 1] + b[n - i, Min[i, n - i]]];
a[n_] := b[n^3 - n, n];
a /@ Range[0, 20] (* Jean-François Alcover, Nov 23 2020, after Alois P. Heinz *)
  • PARI
    {a(n) = polcoeff(prod(k=1, n, 1/(1-x^k+x*O(x^(n^3-n)))), n^3-n)}
  • Python
    import sys
from functools import lru_cache
sys.setrecursionlimit(10**6)
@lru_cache(maxsize=None)
def b(n,i): return 1 if n == 0 or i == 1 else b(n,i-1)+b(n-i,min(i,n-i))
def A304176(n): return b(n**3-n,n) # Chai Wah Wu, Sep 09 2021, after Alois P. Heinz

Formula

a(n) = [x^(n^3-n)] Product_{k=1..n} 1/(1-x^k).

A258297 Number of partitions of n*(n+1)*(n+2) into parts that are at most n.

Original entry on oeis.org

1, 1, 13, 331, 13561, 776594, 57773582, 5320252480, 586352480958, 75438829494131, 11116206652400681, 1848033852642973772, 342436117841931383400, 70020229273505952925559, 15667865938977592230047929, 3809417116914053901413289249, 1000291703885548521424635046427
Offset: 0

Views

Author

Vaclav Kotesovec, May 25 2015

Keywords

Links

Crossrefs

Cf. A238608, A258298, A258299, A258300, A258301.

Programs

  • Maple
    T:=proc(n,k) option remember; `if`(n=0 or k=1, 1, T(n,k-1) + `if`(n

Formula

a(n) ~ exp(2*n + 13/4) * n^(n-3) / (2*Pi).

A258298 Number of partitions of n*(n+1)*(n+2)/6 into parts that are at most n.

Original entry on oeis.org

1, 1, 3, 14, 108, 1115, 14800, 239691, 4602893, 102442041, 2596767156, 73937412122, 2338157235782, 81358388835166, 3090548185022616, 127310130911561966, 5654266354725389764, 269396637045530725099, 13708631585852580662781, 742141584297248778501411
Offset: 0

Views

Author

Vaclav Kotesovec, May 25 2015

Keywords

Links

Crossrefs

Cf. A238608, A258297, A258299, A258300, A258301.

Programs

  • Maple
    T:=proc(n,k) option remember; `if`(n=0 or k=1, 1, T(n,k-1) + `if`(n

Formula

a(n) ~ exp(2*n + 9/2) * n^(n-3) / (2*Pi * 6^(n-1)).

A258299 Number of partitions of n*(n-1)*(n-2) into parts that are at most n.

Original entry on oeis.org

1, 1, 1, 7, 169, 7166, 436140, 34690401, 3418486403, 402588217564, 55217486292383, 8650673262689142, 1524827150449505994, 298774748146352115019, 64436825369109396329518, 15171417879016739747222223, 3872658124805520661780283663, 1065387724298834666633864592587
Offset: 0

Views

Author

Vaclav Kotesovec, May 25 2015

Keywords

Links

Crossrefs

Cf. A238608, A258297, A258298, A258300, A258301.

Programs

  • Maple
    T:=proc(n,k) option remember; `if`(n=0 or k=1, 1, T(n,k-1) + `if`(n

Formula

a(n) ~ exp(2*n - 11/4) * n^(n-3) / (2*Pi).

A258300 Number of partitions of n*(n-1)*(n-2)/6 into parts that are at most n.

Original entry on oeis.org

1, 1, 1, 1, 5, 30, 282, 3539, 55974, 1065947, 23785645, 608889106, 17594781914, 566603884871, 20123663539549, 781500841147604, 32946304088342094, 1498526109256063585, 73147202427442412812, 3814178439827570160925, 211598573411998923138880
Offset: 0

Views

Author

Vaclav Kotesovec, May 25 2015

Keywords

Links

Crossrefs

Cf. A238608, A258297, A258298, A258299, A258301.

Programs

  • Maple
    T:=proc(n,k) option remember; `if`(n=0 or k=1, 1, T(n,k-1) + `if`(n

Formula

a(n) ~ exp(2*n - 3/2) * n^(n-3) / (2*Pi * 6^(n-1)).

A258301 Number of partitions of n*(n+1)*(2n+1)/6 into parts that are at most n.

Original entry on oeis.org

1, 1, 3, 24, 297, 5260, 123755, 3648814, 129828285, 5425234114, 260818130929, 14194798070042, 863357482347465, 58068803644110427, 4281318749672322843, 343463734454952001605, 29792472711307060688049, 2778959190056157071592315, 277420695604265258419161136
Offset: 0

Views

Author

Vaclav Kotesovec, May 25 2015

Keywords

Links

Crossrefs

Cf. A238608, A258297, A258298, A258299, A258300.

Programs

  • Maple
    T:=proc(n,k) option remember; `if`(n=0 or k=1, 1, T(n,k-1) + `if`(n

Formula

a(n) ~ exp(2*n + 9/4) * n^(n-3) / (2*Pi * 3^(n-1)).

A258302 Number of partitions of 2*n^3 into parts that are at most n.

Original entry on oeis.org

1, 1, 9, 271, 16335, 1525940, 196284041, 32409332818, 6561153029810, 1577073620254149, 439541281384464800, 139493983910450106067, 49695878602452933374813, 19646816226938989587513067, 8537966749269377751401117583, 4046350906270352192325991177139
Offset: 0

Views

Author

Vaclav Kotesovec, May 25 2015

Keywords

Links

Crossrefs

Cf. A238608, A258303, A258304, A258305.

Programs

  • Maple
    T:=proc(n,k) option remember; `if`(n=0 or k=1, 1, T(n,k-1) + `if`(n

Formula

a(n) ~ exp(2*n + 1/8) * 2^(n-1) * n^(n-3) / (2*Pi).

A258303 Number of partitions of 3*n^3 into parts that are at most n.

Original entry on oeis.org

1, 1, 13, 588, 53089, 7431069, 1432812535, 354709605775, 107681683621061, 38815870525676822, 16224696168627992214, 7722681288635179285337, 4126484069454572889453794, 2446850787696893234909546422, 1594892857383186062141424302309
Offset: 0

Views

Author

Vaclav Kotesovec, May 25 2015

Keywords

Links

Crossrefs

Cf. A238608, A258302, A258304, A258305.

Programs

  • Maple
    T:=proc(n,k) option remember; `if`(n=0 or k=1, 1, T(n,k-1) + `if`(n

Formula

a(n) ~ exp(2*n + 1/12) * 3^(n-1) * n^(n-3) / (2*Pi).

A258304 Number of partitions of 4*n^3 into parts that are at most n.

Original entry on oeis.org

1, 1, 17, 1027, 123464, 23030612, 5918918145, 1953335236481, 790541795804221, 379916850888632162, 211720519858133280231, 134359691058417334173117, 95719564240981718602134049, 75674822191817499226090337378, 65766024754772296807292428860854
Offset: 0

Views

Author

Vaclav Kotesovec, May 25 2015

Keywords

Links

Crossrefs

Cf. A238608, A258302, A258303, A258305.

Programs

  • Maple
    T:=proc(n,k) option remember; `if`(n=0 or k=1, 1, T(n,k-1) + `if`(n

Formula

a(n) ~ exp(2*n + 1/16) * 4^(n-1) * n^(n-3) / (2*Pi).
Showing 1-10 of 12 results. Next

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