A000412 Number of bipartite partitions of n white objects and 3 black ones.

3, 7, 16, 31, 57, 97, 162, 257, 401, 608, 907, 1325, 1914, 2719, 3824, 5313, 7316, 9973, 13495, 18105, 24132, 31938, 42021, 54948, 71484, 92492, 119120, 152686, 194887, 247693, 313613, 395547, 497154, 622688, 777424, 967525, 1200572, 1485393, 1832779, 2255317

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Number of ways to factor p^n*q^3 where p and q are distinct primes.

Number of Gaussian partitions of n+3*i or 3+n*i where a "Gaussian partition" is a way of writing a Gaussian integer with nonnegative parts as a sum of Gaussian integers with nonnegative parts, imaginary numbers and real numbers. For k = 3+1*i (where i is the imaginary unit), the a(1)=7 ways to write k (where parentheses represent a complex number and a lack of them represents a sum of a real and imaginary number) would be 3+i, (3+i), 2+1+i, (2+i)+1, (1+i)+2, 1+1+1+i, (1+i)+1+1. - Yali Harrary, Nov 20 2022

a(n) is the number of multiset partitions of the multiset {r^n, s^3}. - 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

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..100 from Alois P. Heinz)
• F. C. Auluck, On partitions of bipartite numbers, Proc. Cambridge Philos. Soc. 49, (1953), pp. 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).

Crossrefs

Column 3 of A054225.

Cf. A005380.

Programs

Mathematica

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

Formula

a(n) = if n <= 3 then A054225(3,n), otherwise a(n) = A054225(n,3). - Reinhard Zumkeller, Nov 30 2011

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

a(n) = A000098(n) + A000070(n) + A014153(n). - Yali Harrary, Nov 20 2022

Extensions

Edited by Christian G. Bower, Jan 08 2004