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A000098 Number of partitions of n if there are two kinds of 1, two kinds of 2 and two kinds of 3.
(Formerly M1373 N0533)
11

%I M1373 N0533

%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 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((1-x^k), k>=1)].

%F a(n) = sum(A000097(n-3*j), j=0..floor(n/3)), 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-Fran├žois Alcover_, Feb 04 2016 *)

%t Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@3], {n,0,35}] (* _Robert Price_, Jul 28 2020 *)

%Y Cf. A000070, A008951, A000097, A000710.

%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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Last modified March 3 10:38 EST 2021. Contains 341762 sequences. (Running on oeis4.)