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%I #28 May 30 2021 15:38:16
%S 1,2,5,10,20,36,65,110,185,299,478,744,1147,1732,2591,3817,5573,8036,
%T 11496,16276,22878,31879,44129,60630,82807,112353,151616,203415,
%U 271558,360648,476793,627389,822104,1072668,1394199,1805060,2328653,2993372,3835068,4897199
%N Number of partitions of n into parts but with two kinds of parts of sizes 1 to 8.
%C See A103923 for other combinatorial interpretations of a(n).
%C Also the sum of binomial (D(p), 8) over partitions p of n+36, 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_{k=1..8} 1/(1-x^k))^2*Product_{j>=9} 1/(1-x^j).
%F a(n) = Sum_{j=0..floor(n/8)} A103924(n-8*j), n >= 0.
%F a(n) ~ exp(Pi*sqrt(2*n/3)) * 6^4 * n^3 / (4*sqrt(3) * 8! * Pi^8). - _Vaclav Kotesovec_, Aug 28 2015
%p a:= proc(n) option remember; `if`(n=0, 1, add(add(d+
%p `if`(d<9, d, 0), d=divisors(j))*a(n-j), j=1..n)/n)
%p end:
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Jun 11 2018
%t nmax=60; CoefficientList[Series[Product[1/(1-x^k), {k, 1, 8}] * 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@8], {n,0,39}] (* _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 + 36, 8];
%t Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, May 30 2021 *)
%Y Ninth column (m=8) 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
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