A103929 - OEIS

%I #15 May 30 2021 15:38:27

%S 1,2,5,10,20,36,65,110,185,300,481,751,1162,1762,2647,3918,5748,8331,

%T 11981,17056,24108,33787,47043,65019,89336,121954,165585,223542,

%U 300295,401331,533937,707057,932404,1224376,1601571,2086851,2709449,3505228

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

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

%C 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

%D H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958 (reprinted 1962), p. 91.

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

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

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

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

%t 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 *)

%t Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@10], {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 + 55, 10];

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

%Y 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.

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

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Mar 24 2005