A258668 -id:A258668 - cp's OEIS frontend

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.

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

Alois P. Heinz, Feb 17 2014

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

			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, ...

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.

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.

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 *)

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

A236810 Number of solutions to Sum_{k=1..n} k*c(k) = n! , c(k) >= 0.

Original entry on oeis.org

0, 1, 2, 7, 169, 91606, 2407275335, 4592460368601183, 855163933625625205568537, 20560615981766266405801870502139241, 82864945825700191674729490954631752385038099201, 70899311833745096407560015806403481692583415598602691709750081

Offset: 0

Wouter Meeussen, Feb 08 2014

nonn

a(n) is the number of partitions of n! into parts that are at most n. a(3) = 7: [1,1,1,1,1,1], [2,1,1,1,1], [2,2,1,1], [2,2,2], [3,1,1,1], [3,2,1], [3,3]. - Alois P. Heinz, Feb 08 2014

			for n=3, the 7 solutions are: 3! = 6,0,0 ; 4,1,0 ; 2,2,0 ; 0,3,0 ; 3,0,1 ; 1,1,1 ; 0,0,2.

Alois P. Heinz, Table of n, a(n) for n = 0..31
P. F. Ayuso, J. M. Grau, A. Oller-Marcen, Von Staudt formula for Sum_{z in Z_n[i]} z^k, arXiv preprint arXiv:1402.0333, 2014, Montsh. Math. 178 (2015) 345-359
Vaclav Kotesovec, Graph - the asymptotic ratio (Total 90 terms were computed with a program by Doron Zeilberger)
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], Dec 2011
StackExchange, Combinations sum_{k=1..m} k*n_k = m!, Jan 29 2014

Cf. A008290, A008637, A237512, A258668, A258669, A258670, A258671.

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

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

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

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

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

Vaclav Kotesovec, Jun 07 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..31
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])

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

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

Vaclav Kotesovec, Jun 07 2015

nonn

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.

Vaclav Kotesovec, Table of n, a(n) for n = 0..21
G. J. Rieger, Über Partitionen, Mathematische Annalen (1959), Volume: 138, page 356-362
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])

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

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

Vaclav Kotesovec, Jun 07 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..23
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])

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

a(n) ~ n * (n!)^(2*n-4).

cp's OEIS Frontend

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.

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A236810 Number of solutions to Sum_{k=1..n} k*c(k) = n! , c(k) >= 0.

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

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

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

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