Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I M4964 N2129 #42 Jan 01 2024 08:01:21
%S 15,45,118,257,522,975,1752,2998,4987,8043,12693,19584,29719,44324,
%T 65210,94642,135805,192699,270822,377048,520624,713123,969784,1309646,
%U 1757447,2343931,3108553,4100220,5380964,7027376,9135769,11824507
%N Number of bipartite partitions of n white objects and 7 black ones.
%C Number of ways to factor p^n*q^7 where p and q are distinct primes.
%C a(n) is the number of multiset partitions of the multiset {r^n, s^7}. - _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. 1.
%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="/A002756/b002756.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 <= 7 then A054225(7,n) else A054225(n,7). - _Reinhard Zumkeller_, Nov 30 2011
%F a(n) ~ 3*n^(5/2) * exp(Pi*sqrt(2*n/3)) / (140*sqrt(2)*Pi^7). - _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^7, p^n*q^7]; Table[a[n], {n, 0, 31}] (* _Jean-François Alcover_, Mar 17 2014, after _Alois P. Heinz_ *)
%t nmax = 50; CoefficientList[Series[(15 + 15*x + 13*x^2 + 6*x^3 - 5*x^4 - 15*x^5 - 28*x^6 - 34*x^7 - 26*x^8 - 10*x^9 + 6*x^10 + 25*x^11 + 27*x^12 + 31*x^13 + 20*x^14 + 3*x^15 - 9*x^16 - 16*x^17 - 17*x^18 - 9*x^19 - 4*x^20 + 8*x^22 + 6*x^23 + 4*x^24 - 3*x^25 - 3*x^26 + x^27)/((1-x) * (1-x^2) * (1-x^3) * (1-x^4) * (1-x^5) * (1-x^6) * (1-x^7)) * Product[1/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Feb 01 2016 *)
%Y Column 7 of A054225.
%Y Cf. A005380.
%K nonn
%O 0,1
%A _N. J. A. Sloane_
%E Edited by _Christian G. Bower_, Jan 08 2004