A238608 -id:A238608

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.

+10
27

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,9

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

LINKS

Alois P. Heinz, Antidiagonals n = 0..54, flattened

A. V. Sills and D. Zeilberger, Formulae for the number of partitions of n into at most m parts (using the quasi-polynomial ansatz) arXiv:1108.4391 [math.CO], 2011.

FORMULA

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

EXAMPLE

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, ...

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 *)

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.

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Feb 17 2014

STATUS

approved

A304176

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Chai Wah Wu, Table of n, a(n) for n = 0..136 (terms 0..96 from Alois P. Heinz)

FORMULA

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

EXAMPLE

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.

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 *)

PROG

(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

CROSSREFS

Cf. A206240, A238608, A304208, A304212.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, May 07 2018

STATUS

approved

A258297

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..134

FORMULA

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

MAPLE

T:=proc(n, k) option remember; `if`(n=0 or k=1, 1, T(n, k-1) + `if`(n<k, 0, T(n-k, k))) end proc: seq(T(n*(n+1)*(n+2), n), n=0..20);

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, May 25 2015

STATUS

approved

A258298

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..135

FORMULA

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

MAPLE

T:=proc(n, k) option remember; `if`(n=0 or k=1, 1, T(n, k-1) + `if`(n<k, 0, T(n-k, k))) end proc: seq(T(n*(n+1)*(n+2)/6, n), n=0..20);

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, May 25 2015

STATUS

approved

A258299

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..89

FORMULA

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

MAPLE

T:=proc(n, k) option remember; `if`(n=0 or k=1, 1, T(n, k-1) + `if`(n<k, 0, T(n-k, k))) end proc: seq(T(n*(n-1)*(n-2), n), n=0..20);

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, May 25 2015

STATUS

approved

A258300

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..141

FORMULA

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

MAPLE

T:=proc(n, k) option remember; `if`(n=0 or k=1, 1, T(n, k-1) + `if`(n<k, 0, T(n-k, k))) end proc: seq(T(n*(n-1)*(n-2)/6, n), n=0..20);

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, May 25 2015

STATUS

approved

A258301

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..118

FORMULA

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

MAPLE

T:=proc(n, k) option remember; `if`(n=0 or k=1, 1, T(n, k-1) + `if`(n<k, 0, T(n-k, k))) end proc: seq(T(n*(n+1)*(2n+1)/6, n), n=0..20);

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, May 25 2015

STATUS

approved

A258302

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..114

FORMULA

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

MAPLE

T:=proc(n, k) option remember; `if`(n=0 or k=1, 1, T(n, k-1) + `if`(n<k, 0, T(n-k, k))) end proc: seq(T(2*n^3, n), n=0..20);

CROSSREFS

Cf. A238608, A258303, A258304, A258305.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, May 25 2015

STATUS

approved

A258303

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..100

FORMULA

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

MAPLE

T:=proc(n, k) option remember; `if`(n=0 or k=1, 1, T(n, k-1) + `if`(n<k, 0, T(n-k, k))) end proc: seq(T(3*n^3, n), n=0..20);

CROSSREFS

Cf. A238608, A258302, A258304, A258305.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, May 25 2015

STATUS

approved

A258304

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..62

FORMULA

a(n) ~ exp(2*n + 1/16) * 4^(n-1) * n^(n-3) / (2*Pi).

MAPLE

T:=proc(n, k) option remember; `if`(n=0 or k=1, 1, T(n, k-1) + `if`(n<k, 0, T(n-k, k))) end proc: seq(T(4*n^3, n), n=0..20);

CROSSREFS

Cf. A238608, A258302, A258303, A258305.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, May 25 2015

STATUS

approved