A002758 Number of bipartite partitions of n white objects and 9 black ones. (Formerly M5209 N2265)

30, 97, 267, 608, 1279, 2472, 4571, 8043, 13715, 22652, 36535, 57568, 89079, 135384, 202747, 299344, 436597, 629364, 897970, 1268634, 1776562, 2466961, 3399463, 4650218, 6318429, 8529869, 11446563, 15272827, 20269135, 26762094

Offset

0,1

Comments

Number of ways to factor p^n*q^9 where p and q are distinct primes.

a(n) is the number of multiset partitions of the multiset {r^n, s^9}. - 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. 2.

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..1000

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 <= 9 then A054225(9,n) else A054225(n,9). - Reinhard Zumkeller, Nov 30 2011

a(n) ~ n^(7/2) * exp(Pi*sqrt(2*n/3)) / (560*sqrt(2)*Pi^9). - 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^9, p^n*q^9]; Table[a[n], {n, 0, 29}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)
nmax = 50; CoefficientList[Series[(30 + 37*x + 43*x^2 + 37*x^3 + 20*x^4 - 3*x^5 - 35*x^6 - 65*x^7 - 97*x^8 - 119*x^9 - 109*x^10 - 69*x^11 - 26*x^12 + 37*x^13 + 89*x^14 + 131*x^15 + 142*x^16 + 141*x^17 + 97*x^18 + 44*x^19 - 18*x^20 - 72*x^21 - 100*x^22 - 108*x^23 - 96*x^24 - 69*x^25 - 25*x^26 + 12*x^27 + 42*x^28 + 52*x^29 + 54*x^30 + 35*x^31 + 14*x^32 + 2*x^33 - 4*x^34 - 20*x^35 - 19*x^36 - 14*x^37 - 8*x^38 + 7*x^39 + 8*x^40 + 8*x^41 - 2*x^42 - 4*x^43 + x^44)/((1-x) * (1-x^2) * (1-x^3) * (1-x^4) * (1-x^5) * (1-x^6) * (1-x^7) * (1-x^8) * (1-x^9)) * Product[1/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 01 2016 *)

Crossrefs

Column 9 of A054225.

Cf. A005380.

Sequence in context: A308508 A096382 A303859 * A346855 A187401 A130863

Adjacent sequences: A002755 A002756 A002757 * A002759 A002760 A002761

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Edited by Christian G. Bower, Jan 08 2004

Status

approved