cp's OEIS Frontend

A103925 Number of partitions of n into parts but with two kinds of parts of sizes 1,2,3,4,5 and 6.

%I A103925 #29 May 30 2021 15:38:02
%S A103925 1,2,5,10,20,36,65,109,182,292,463,714,1091,1631,2416,3523,5091,7264,
%T A103925 10284,14405,20035,27621,37831,51425,69497,93299,124588,165408,218533,
%U A103925 287231,375851,489525,634980,820195,1055444,1352965,1728326,2200060,2791516,3530513
%N A103925 Number of partitions of n into parts but with two kinds of parts of sizes 1,2,3,4,5 and 6.
%C A103925 See A103923 for other combinatorial interpretations of a(n).
%C A103925 Also the sum of binomial (D(p), 6) over partitions p of n+21, where D(p) is the number of different part sizes in p. - _Emily Anible_, Jun 09 2018
%D A103925 H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958 (reprinted 1962), p. 90.
%D A103925 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.
%H A103925 Alois P. Heinz and Vaclav Kotesovec, <a href="/A103925/b103925.txt">Table of n, a(n) for n = 0..5000</a> (first 1000 terms from Alois P. Heinz)
%F A103925 G.f.: (product(1/(1-x^k), k=1..6)^2)*product(1/(1-x^j), j=7..infty).
%F A103925 a(n) = sum(A103924(n-6*j), j=0..floor(n/6)), n>=0.
%F A103925 a(n) ~ exp(Pi*sqrt(2*n/3)) * 6^3 * n^2 / (4*sqrt(3) * 6! * Pi^6) = exp(Pi*sqrt(2*n/3)) * sqrt(3) * n^2 / (40*Pi^6). - _Vaclav Kotesovec_, Aug 28 2015
%p A103925 with(numtheory): a:= proc(n) option remember; `if`(n=0, 1, add(add(d*`if`(d<7, 2, 1), d=divisors(j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=0..40);  # _Alois P. Heinz_, Sep 14 2014
%t A103925 nmax=60; CoefficientList[Series[Product[1/(1-x^k), {k, 1, 6}] * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 28 2015 *)
%t A103925 Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@6],  {n,0,39}] (* _Robert Price_, Jul 29 2020 *)
%t A103925 T[n_, 0] := PartitionsP[n];
%t A103925 T[n_, m_] /; (n >= m(m+1)/2) := T[n, m] = T[n-m, m-1] + T[n-m, m];
%t A103925 T[_, _] = 0;
%t A103925 a[n_] := T[n+21, 6];
%t A103925 Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, May 30 2021 *)
%Y A103925 Seventh column (m=6) of Fine-Riordan triangle A008951, of triangle A103923, i.e. the p_2(n, m) array of the Gupta et al. reference.
%Y A103925 Cf. A000712 (all parts of two kinds).
%K A103925 nonn,easy
%O A103925 0,2
%A A103925 _Wolfdieter Lang_, Mar 24 2005