|
|
A002757
|
|
Number of bipartite partitions of n white objects and 8 black ones.
(Formerly M5123 N2219)
|
|
4
|
|
|
22, 67, 181, 401, 831, 1576, 2876, 4987, 8406, 13715, 21893, 34134, 52327, 78785, 116982, 171259, 247826, 354482, 502090, 704265, 979528, 1351109, 1849932, 2514723, 3396262, 4557867, 6081466, 8068930, 10650479, 13987419, 18283999
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,1
|
|
COMMENTS
|
Number of ways to factor p^n*q^8 where p and q are distinct primes.
a(n) is the number of multiset partitions of the multiset {r^n, s^8}. - Joerg Arndt, Jan 01 2024
|
|
REFERENCES
|
M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956, p. 1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
LINKS
|
|
|
FORMULA
|
a(n) ~ 3*sqrt(3) * n^3 * exp(Pi*sqrt(2*n/3)) / (1120*Pi^8). - Vaclav Kotesovec, Feb 01 2016
|
|
MATHEMATICA
|
p = 2; q = 3; b[n_, k_] := b[n, k] = If[n>k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d>k, 0, b[n/d, d]], {d, DeleteCases[Divisors[n], 1|n]}]]; a[n_] := b[p^n*q^8, p^n*q^8]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)
nmax = 50; CoefficientList[Series[(22 + 23*x + 25*x^2 + 16*x^3 + 4*x^4 - 14*x^5 - 34*x^6 - 50*x^7 - 65*x^8 - 52*x^9 - 32*x^10 + 5*x^11 + 27*x^12 + 57*x^13 + 67*x^14 + 65*x^15 + 42*x^16 + 15*x^17 - 14*x^18 - 34*x^19 - 40*x^20 - 46*x^21 - 26*x^22 - 8*x^23 + 8*x^24 + 11*x^25 + 18*x^26 + 14*x^27 + 9*x^28 + 3*x^29 - 7*x^30 - 7*x^31 - 6*x^32 + 3*x^33 + 3*x^34 - x^35)/((1-x) * (1-x^2) * (1-x^3) * (1-x^4) * (1-x^5) * (1-x^6) * (1-x^7) * (1-x^8)) * Product[1/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 01 2016 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|