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%I M1373 N0533 #51 Feb 16 2022 05:32:28
%S 1,2,5,10,19,33,57,92,147,227,345,512,752,1083,1545,2174,3031,4179,
%T 5719,7752,10438,13946,18519,24428,32051,41805,54265,70079,90102,
%U 115318,147005,186626,236064,297492,373645,467707
%N Number of partitions of n if there are two kinds of 1, two kinds of 2 and two kinds of 3.
%C Also number of partitions of 2*n+1 with exactly 3 odd parts (offset 1). - _Vladeta Jovovic_, Jan 12 2005
%C Convolution of A000041 and A001399. - _Vaclav Kotesovec_, Aug 18 2015
%C Also the sum of binomial(D(p),3) over partitions p of n+6, where D(p) is the number of different sizes of parts in p. - _Emily Anible_, May 13 2018
%D H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 90.
%D J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A000098/b000098.txt">Table of n, a(n) for n = 0..1000</a>
%H Álvaro Gutiérrez and Mercedes H. Rosas, <a href="https://arxiv.org/abs/2201.00240">Partial symmetries of iterated plethysms</a>, arXiv:2201.00240 [math.CO], 2022.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F Euler transform of 2 2 2 1 1 1 1...
%F G.f.: 1/((1-x)(1-x^2)(1-x^3)*Product_{k>=1} (1-x^k)).
%F a(n) = Sum_{j=0..floor(n/3)} A000097(n-3*j), n >= 0.
%F a(n) ~ sqrt(n) * exp(Pi*sqrt(2*n/3)) / (2*sqrt(2)*Pi^3). - _Vaclav Kotesovec_, Aug 18 2015
%e a(3)=10 because we have 3, 3', 2+1, 2+1', 2'+1, 2'+1', 1+1+1, 1+1+1', 1+1'+1' and 1'+1'+1'.
%t CoefficientList[1/((1-x)*(1-x^2)*(1-x^3)*QPochhammer[x]) + O[x]^40, x] (* _Jean-François Alcover_, Feb 04 2016 *)
%t Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@3], {n,0,35}] (* _Robert Price_, Jul 28 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 + 6, 3];
%t Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, May 30 2021 *)
%Y Cf. A000070, A008951, A000097, A000710.
%Y Cf. A000041, A001399.
%Y Fourth column of Riordan triangle A008951 and of triangle A103923.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E Edited by _Emeric Deutsch_, Mar 23 2005
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