%I M5209 N2265 #41 Jan 01 2024 08:01:11
%S 30,97,267,608,1279,2472,4571,8043,13715,22652,36535,57568,89079,
%T 135384,202747,299344,436597,629364,897970,1268634,1776562,2466961,
%U 3399463,4650218,6318429,8529869,11446563,15272827,20269135,26762094
%N Number of bipartite partitions of n white objects and 9 black ones.
%C Number of ways to factor p^n*q^9 where p and q are distinct primes.
%C a(n) is the number of multiset partitions of the multiset {r^n, s^9}. - _Joerg Arndt_, Jan 01 2024
%D 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.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A002758/b002758.txt">Table of n, a(n) for n = 0..1000</a>
%H M. S. Cheema and H. Gupta, <a href="/A002755/a002755.pdf">Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956</a> (Annotated scanned pages from, plus a review)
%F a(n) = if n <= 9 then A054225(9,n) else A054225(n,9). - _Reinhard Zumkeller_, Nov 30 2011
%F a(n) ~ n^(7/2) * exp(Pi*sqrt(2*n/3)) / (560*sqrt(2)*Pi^9). - _Vaclav Kotesovec_, Feb 01 2016
%t 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_ *)
%t 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 *)
%Y Column 9 of A054225.
%Y Cf. A005380.
%K nonn
%O 0,1
%A _N. J. A. Sloane_
%E Edited by _Christian G. Bower_, Jan 08 2004