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

0,2

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.

Links

Table of n, a(n) for n=0..39.

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

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 *)

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).

Sequence in context: A103925 A160525 A103926 * A103928 A103929 A121597

Adjacent sequences: A103924 A103925 A103926 * A103928 A103929 A103930

Keyword

nonn,easy

Author

Wolfdieter Lang, Mar 24 2005

Status

approved