A000465 Number of bipartite partitions of n white objects and 4 black ones. (Formerly M3821 N1565)

5, 12, 29, 57, 109, 189, 323, 522, 831, 1279, 1941, 2876, 4215, 6066, 8644, 12151, 16933, 23336, 31921, 43264, 58250, 77825, 103362, 136371, 178975, 233532, 303268, 391831, 504069, 645520, 823419, 1046067, 1324136, 1669950, 2099104, 2629685, 3284325, 4089300

Offset

0,1

Comments

Number of ways to factor p^n*q^4 where p and q are distinct primes.

a(n) is the number of multiset partitions of the multiset {r^n, s^4}. - 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

Alois P. Heinz, Table of n, a(n) for n = 0..5000

F. C. Auluck, On partitions of bipartite numbers, Proc. Cambridge Philos. Soc. 49, (1953). 72-83.

F. C. Auluck, On partitions of bipartite numbers, annotated scan of a few pages.

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 <= 4 then A054225(4,n) else A054225(n,4). - Reinhard Zumkeller, Nov 30 2011

a(n) ~ sqrt(3) * n * exp(Pi*sqrt(2*n/3)) / (8*Pi^4). - Vaclav Kotesovec, Feb 01 2016

Mathematica

max = 40; col = 4; 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[(5 + 2*x - 3*x^3 - 5*x^4 - x^5 + 3*x^7 + x^8 - x^9)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) * Product[1/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 01 2016 *)

Crossrefs

Column 4 of A054225.

Cf. A005380.

Sequence in context: A002767 A055245 A196410 * A283506 A266471 A069306

Adjacent sequences: A000462 A000463 A000464 * A000466 A000467 A000468

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Edited by Christian G. Bower, Jan 08 2004

Status

approved