A103927 Number of partitions of n into parts but with two kinds of parts of sizes 1 to 8.

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

Offset: 0

Author

Wolfdieter Lang, Mar 24 2005

Keywords

• nonn
• easy

Comments

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

References

• 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.

Crossrefs

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).

Programs

Maple

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:
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 11 2018

Mathematica

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];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 30 2021 *)

Formula

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