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A103924 Number of partitions of n into parts but with two kinds of parts of sizes 1,2,3,4 and 5.

%I A103924 #33 May 30 2021 12:21:31
%S A103924 1,2,5,10,20,36,64,107,177,282,443,678,1026,1522,2234,3231,4628,6550,
%T A103924 9193,12774,17619,24098,32740,44161,59213,78894,104553,137787,180702,
%U A103924 235806,306354,396226,510392,654787,836911,1065734,1352475,1710535,2156536,2710318
%N A103924 Number of partitions of n into parts but with two kinds of parts of sizes 1,2,3,4 and 5.
%C A103924 See A103923 for other combinatorial interpretations of a(n).
%C A103924 Convolution of A001401 and A000041. - _Vaclav Kotesovec_, Aug 28 2015
%C A103924 Also the sum of binomial (D(p), 5) over partitions p of n+15, where D(p) is the number of different part sizes in p. - _Emily Anible_, Jun 09 2018
%D A103924 H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958 (reprinted 1962), p. 90.
%D A103924 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.
%H A103924 Alois P. Heinz, <a href="/A103924/b103924.txt">Table of n, a(n) for n = 0..10000</a>
%F A103924 G.f.: (product(1/(1-x^k), k=1..5)^2)*product(1/(1-x^j), j=6..infty).
%F A103924 a(n) = sum(A000710(n-5*j), j=0..floor(n/5)), n>=0.
%F A103924 a(n) ~ 3*n^(3/2) * exp(Pi*sqrt(2*n/3)) / (20*sqrt(2)*Pi^5). - _Vaclav Kotesovec_, Aug 28 2015
%p A103924 with(numtheory): a:= proc(n) a(n):=`if`(n=0, 1, add(add(d*`if`(d<6, 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 A103924 a[n_] := a[n] = If[n==0, 1, Sum[Sum[d*If[d<6, 2, 1], {d, Divisors[j]}] * a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Aug 28 2015, after _Alois P. Heinz_ *)
%t A103924 nmax=60; CoefficientList[Series[Product[1/(1-x^k), {k, 1, 5}] * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 28 2015 *)
%t A103924 Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@5],  {n,0,39}] (* _Robert Price_, Jul 29 2020 *)
%t A103924 T[n_, 0] := PartitionsP[n];
%t A103924 T[n_, m_] /; (n >= m(m+1)/2) := T[n, m] = T[n-m, m-1] + T[n-m, m];
%t A103924 T[_, _] = 0;
%t A103924 a[n_] := T[n+15, 5];
%t A103924 Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, May 30 2021 *)
%Y A103924 Sixth column (m=5) of Fine-Riordan triangle A008951 and of triangle A103923, i.e. the p_2(n, m) array of the Gupta et al. reference.
%Y A103924 Cf. A000712 (all parts of two kinds).
%K A103924 nonn,easy
%O A103924 0,2
%A A103924 _Wolfdieter Lang_, Mar 24 2005