A103929 Number of partitions of n into parts but with two kinds of parts of sizes 1 to 10.

1, 2, 5, 10, 20, 36, 65, 110, 185, 300, 481, 751, 1162, 1762, 2647, 3918, 5748, 8331, 11981, 17056, 24108, 33787, 47043, 65019, 89336, 121954, 165585, 223542, 300295, 401331, 533937, 707057, 932404, 1224376, 1601571, 2086851, 2709449, 3505228

Offset: 0

Author

Wolfdieter Lang, Mar 24 2005

Keywords

• nonn
• easy

Comments

See A103923 for other combinatorial interpretations of a(n).

In general, column m of A008951 is asymptotic to exp(Pi*sqrt(2*n/3)) * 6^(m/2) * n^((m-2)/2) / (4*sqrt(3) * m! * Pi^m), equivalently to 6^(m/2) * n^(m/2) / (m! * Pi^m) * p(n), where p(n) is the partition function A000041. - Vaclav Kotesovec, Aug 28 2015

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

Eleventh column (m=10) 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, 10}] * 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@10], {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 + 55, 10];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 30 2021 *)

Formula

G.f.: (product(1/(1-x^k), k=1..10)^2)*product(1/(1-x^j), j=11..infty).

a(n)=sum(A103924(n-10*j), j=0..floor(n/10)), n>=0.

a(n) ~ exp(Pi*sqrt(2*n/3)) * 6^5 * n^4 / (4*sqrt(3) * 10! * Pi^10). - Vaclav Kotesovec, Aug 28 2015