A103928 Number of partitions of n into parts but with two kinds of parts of sizes 1 to 9.

1, 2, 5, 10, 20, 36, 65, 110, 185, 300, 480, 749, 1157, 1752, 2627, 3882, 5683, 8221, 11796, 16756, 23627, 33036, 45881, 63257, 86689, 118036, 159837, 215211, 288314, 384275, 509829, 673270, 885361, 1159357, 1512235, 1964897, 2543864, 3281686

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), 9) over partitions p of n+45, 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. 91.
• J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.

Crossrefs

Tenth column (m=9) 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

Mathematica

nmax=60; CoefficientList[Series[Product[1/(1-x^k), {k, 1, 9}] * 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@9], {n,0,37}] (* 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 + 45, 9];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 30 2021 *)

Formula

G.f.: (Product_{k=1..9} 1/(1-x^k))^2 * Product_{j>=10} 1/(1-x^j).

a(n) = Sum_{j=0..floor(n/9)} A103924(n-9*j), n >= 0.

a(n) ~ exp(Pi*sqrt(2*n/3)) * 6^(9/2) * n^(7/2) / (4*sqrt(3) * 9! * Pi^9). - Vaclav Kotesovec, Aug 28 2015