A258304 -id:A258304

Search: a258304 -id:a258304

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A238608

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

+10
13

1, 1, 5, 75, 2280, 106852, 6889527, 569704489, 57733506640, 6944433285769, 968356321790171, 153738253618009045, 27396489338187214000, 5417302365503826145732, 1177436831956414016252071, 279074576444362385794783853, 71649589941044468875380333533

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

OFFSET

0,3

COMMENTS

In general, "number of partitions of j*n^3 into parts that are at most n" is (for j>0) asymptotic to exp(2*n + 1/(4*j)) * n^(n-3) * j^(n-1) / (2*Pi). - Vaclav Kotesovec, May 25 2015

LINKS

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..122 (terms 0..70 from Alois P. Heinz)

FORMULA

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

a(n) ~ exp(2*n + 1/4) * n^(n-3) / (2*Pi). - Vaclav Kotesovec, May 25 2015

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^3, n), n=0..20); # Vaclav Kotesovec, May 25 2015 after Alois P. Heinz

MATHEMATICA

a[n_] := SeriesCoefficient[1/QPochhammer[q, q, n], {q, 0, n^3}]; Table[ a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 03 2015 *)

CROSSREFS

Column k=3 of A238016.

Cf. A258302 (j=2), A258303 (j=3), A258304 (j=4), A258305 (j=5).

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Mar 01 2014

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

A258305

Number of partitions of 5*n^3 into parts that are at most n.

+10
5

1, 1, 21, 1587, 238383, 55567352, 17847892852, 7361757422695, 3723968532118769, 2236948326023829383, 1558198571940473783110, 1236019919143994867274825, 1100668944858994534988670451, 1087699749857592852109688615310, 1181577954513871365541825872100466

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

OFFSET

0,3

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..90

FORMULA

a(n) ~ exp(2*n + 1/20) * 5^(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(5*n^3, n), n=0..20);

CROSSREFS

Cf. A238608, A258302, A258303, A258304.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, May 25 2015

STATUS

approved

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