A002756 Number of bipartite partitions of n white objects and 7 black ones. (Formerly M4964 N2129)

15, 45, 118, 257, 522, 975, 1752, 2998, 4987, 8043, 12693, 19584, 29719, 44324, 65210, 94642, 135805, 192699, 270822, 377048, 520624, 713123, 969784, 1309646, 1757447, 2343931, 3108553, 4100220, 5380964, 7027376, 9135769, 11824507

Offset

0,1

Comments

Number of ways to factor p^n*q^7 where p and q are distinct primes.

a(n) is the number of multiset partitions of the multiset {r^n, s^7}. - 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..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 <= 7 then A054225(7,n) else A054225(n,7). - Reinhard Zumkeller, Nov 30 2011

a(n) ~ 3*n^(5/2) * exp(Pi*sqrt(2*n/3)) / (140*sqrt(2)*Pi^7). - 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^7, p^n*q^7]; Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)
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 *)

Crossrefs

Column 7 of A054225.

Cf. A005380.

Sequence in context: A270563 A126228 A072251 * A039450 A127069 A346853

Adjacent sequences: A002753 A002754 A002755 * A002757 A002758 A002759

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Edited by Christian G. Bower, Jan 08 2004

Status

approved