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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
Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Alois P. Heinz)
F. C. Auluck, On partitions of bipartite numbers, Proc. Cambridge Philos. Soc. 49, (1953). 72-83.
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. (Annotated scanned pages from, plus a review)
FORMULA
a(n) = if n <= 2 then A054225(2,n) else A054225(n,2). - Reinhard Zumkeller, Nov 30 2011
From Vaclav Kotesovec, Feb 01 2016, corrected Nov 05 2016: (Start)
a(n) = A000070(n) + A000097(n).
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
Column 2 of A054225.
Cf. A005380.
Sequence in context: A090676 A333779 A261240 * A081055 A034446 A174511
KEYWORD
nonn
AUTHOR
EXTENSIONS
Edited by Christian G. Bower, Jan 08 2004
STATUS
approved

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)