A103926 Number of partitions of n into parts but with two kinds of parts of sizes 1 to 7.

1, 2, 5, 10, 20, 36, 65, 110, 184, 297, 473, 734, 1127, 1696, 2526, 3707, 5388, 7737, 11018, 15532, 21731, 30147, 41538, 56813, 77234, 104317, 140120, 187139, 248680, 328769, 432664, 566759, 739297, 960315, 1242583, 1601645, 2057095, 2632724

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), 7) over partitions p of n+28, 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

Eighth column (m=7) 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, 7}] * 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@7], {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 + 28, 7];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 30 2021 *)

Formula

G.f.: (product(1/(1-x^k), k=1..7)^2)*product(1/(1-x^j), j=8..infty).

a(n) = sum(A103924(n-7*j), j=0..floor(n/7)), n>=0.

a(n) ~ exp(Pi*sqrt(2*n/3)) * 6^(7/2) * n^(5/2) / (4*sqrt(3) * 7! * Pi^7). - Vaclav Kotesovec, Aug 28 2015