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
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
-
Table[Coefficient[Series[Product[1/(1- x^k),{k,n}],{x,0,n!}],x^(n!)] ,{n,7}]
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
a(1) = 1: 1.
a(2) = 3: 22, 211, 1111.
a(3) = 75: 333333333, ..., 111111111111111111111111111.
-
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 *)
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
a(1) = 1: 11.
a(2) = 3: 22, 211, 1111.
a(3) = 10: 332, 2222, 3221, 3311, 22211, 32111, 221111, 311111, 2111111, 11111111.
-
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 *)
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
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.
-
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 *)
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
a(1) = 1: 1.
a(2) = 1: 211.
a(3) = 48: 3333333321, ..., 321111111111111111111111.
-
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 *)
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