A103926 - OEIS

%I #24 May 30 2021 15:38:11

%S 1,2,5,10,20,36,65,110,184,297,473,734,1127,1696,2526,3707,5388,7737,

%T 11018,15532,21731,30147,41538,56813,77234,104317,140120,187139,

%U 248680,328769,432664,566759,739297,960315,1242583,1601645,2057095,2632724

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

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

%C 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

%D H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958 (reprinted 1962), p. 90.

%D J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.

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

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

%F 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

%t 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 *)

%t Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@7], {n,0,37}] (* _Robert Price_, Jul 29 2020 *)

%t T[n_, 0] := PartitionsP[n];

%t T[n_, m_] /; (n >= m (m + 1)/2) := T[n, m] = T[n - m, m - 1] + T[n - m, m];

%t T[_, _] = 0;

%t a[n_] := T[n + 28, 7];

%t Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, May 30 2021 *)

%Y 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.

%Y Cf. A000712 (all parts of two kinds).

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Mar 24 2005