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A103927
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Number of partitions of n into parts but with two kinds of parts of sizes 1 to 8.
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0
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1, 2, 5, 10, 20, 36, 65, 110, 185, 299, 478, 744, 1147, 1732, 2591, 3817, 5573, 8036, 11496, 16276, 22878, 31879, 44129, 60630, 82807, 112353, 151616, 203415, 271558, 360648, 476793, 627389, 822104, 1072668, 1394199, 1805060, 2328653, 2993372, 3835068, 4897199
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OFFSET
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0,2
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COMMENTS
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See A103923 for other combinatorial interpretations of a(n).
Also the sum of binomial (D(p), 8) over partitions p of n+36, where D(p) is the number of different part sizes in p. - Emily Anible, Jun 09 2018
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REFERENCES
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H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958 (reprinted 1962), p. 90.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.
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LINKS
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FORMULA
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G.f.: (Product_{k=1..8} 1/(1-x^k))^2*Product_{j>=9} 1/(1-x^j).
a(n) = Sum_{j=0..floor(n/8)} A103924(n-8*j), n >= 0.
a(n) ~ exp(Pi*sqrt(2*n/3)) * 6^4 * n^3 / (4*sqrt(3) * 8! * Pi^8). - Vaclav Kotesovec, Aug 28 2015
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MAPLE
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a:= proc(n) option remember; `if`(n=0, 1, add(add(d+
`if`(d<9, d, 0), d=divisors(j))*a(n-j), j=1..n)/n)
end:
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MATHEMATICA
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nmax=60; CoefficientList[Series[Product[1/(1-x^k), {k, 1, 8}] * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 28 2015 *)
Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@8], {n, 0, 39}] (* Robert Price, Jul 29 2020 *)
T[n_, 0] := PartitionsP[n];
T[n_, m_] /; (n >= m (m + 1)/2) := T[n, m] = T[n - m, m - 1] + T[n - m, m];
T[_, _] = 0;
a[n_] := T[n + 36, 8];
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CROSSREFS
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Ninth column (m=8) of Fine-Riordan triangle A008951 and of triangle A103923, i.e., the p_2(n, m) array of the Gupta et al. reference.
Cf. A000712 (all parts of two kinds).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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