|
|
A000291
|
|
Number of bipartite partitions of n white objects and 2 black ones.
(Formerly M1168 N0447)
|
|
7
|
|
|
2, 4, 9, 16, 29, 47, 77, 118, 181, 267, 392, 560, 797, 1111, 1541, 2106, 2863, 3846, 5142, 6808, 8973, 11733, 15275, 19753, 25443, 32582, 41569, 52770, 66757, 84078, 105555, 131995, 164566, 204450, 253292, 312799, 385285, 473183, 579722, 708353, 863553
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,1
|
|
COMMENTS
|
Number of ways to factor p^n*q^2 where p and q are distinct primes.
a(n) is the number of multiset partitions of the multiset {r^n, s^2}. - 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.
Amarnath Murthy, "Generalization of Smarandache Factor Partition introducing Smarandache Factor Partition". Smarandache Notions Journal, 1-2-3, vol. 11, 2000.
Amarnath Murthy, Program for finding out the number of Smarandache Factor Partitions. Smarandache Notions Journal, Vol. 13, 2002.
Amarnath Murthy, e-book, MS LIT format, "Ideas on Smarandache Notions".
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.9, 1.14.
D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778. - N. J. A. Sloane, Dec 30 2018
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) ~ sqrt(3) * exp(Pi*sqrt(2*n/3)) / (4*Pi^2) * (1 + 83*Pi/(24*sqrt(6*n))).
(End)
|
|
EXAMPLE
|
a(2) = 9: let p = 2 and q = 3, p^2*q^2 = 36; there are 9 factorizations: (36), (18*2), (12*3), (9*4), (9*2^2), (6*6), (6*3*2), (4*3^2), (3^2*2^2).
|
|
MATHEMATICA
|
max = 40; col = 2; s1 = Series[Product[1/(1-x^(n-k)*y^k), {n, 1, max+2}, {k, 0, n}], {y, 0, col}] // Normal; s2 = Series[s1, {x, 0, max+1}]; a[n_] := SeriesCoefficient[s2, {x, 0, n}, {y, 0, col}]; Table[ a[n] , {n, 0, max}] (* Jean-François Alcover, Mar 13 2014 *)
nmax = 50; CoefficientList[Series[1/(1-x)*(1 + 1/(1-x^2))*Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 01 2016 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|