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A002756
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Number of bipartite partitions of n white objects and 7 black ones.
(Formerly M4964 N2129)
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5
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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
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OFFSET
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0,1
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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FORMULA
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a(n) ~ 3*n^(5/2) * exp(Pi*sqrt(2*n/3)) / (140*sqrt(2)*Pi^7). - Vaclav Kotesovec, Feb 01 2016
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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