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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) 12
 1, 2, 5, 10, 19, 33, 57, 92, 147, 227, 345, 512, 752, 1083, 1545, 2174, 3031, 4179, 5719, 7752, 10438, 13946, 18519, 24428, 32051, 41805, 54265, 70079, 90102, 115318, 147005, 186626, 236064, 297492, 373645, 467707 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Also number of partitions of 2*n+1 with exactly 3 odd parts (offset 1). - Vladeta Jovovic, Jan 12 2005 Convolution of A000041 and A001399. - Vaclav Kotesovec, Aug 18 2015 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 REFERENCES H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 90. J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n = 0..1000 Álvaro Gutiérrez and Mercedes H. Rosas, Partial symmetries of iterated plethysms, arXiv:2201.00240 [math.CO], 2022. N. J. A. Sloane, Transforms FORMULA Euler transform of 2 2 2 1 1 1 1... G.f.: 1/((1-x)(1-x^2)(1-x^3)*Product_{k>=1} (1-x^k)). a(n) = Sum_{j=0..floor(n/3)} A000097(n-3*j), n >= 0. a(n) ~ sqrt(n) * exp(Pi*sqrt(2*n/3)) / (2*sqrt(2)*Pi^3). - Vaclav Kotesovec, Aug 18 2015 EXAMPLE 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'. MATHEMATICA CoefficientList[1/((1-x)*(1-x^2)*(1-x^3)*QPochhammer[x]) + O[x]^40, x] (* Jean-François Alcover, Feb 04 2016 *) Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@3], {n, 0, 35}] (* Robert Price, Jul 28 2020 *) T[n_, 0] := PartitionsP[n]; T[n_, m_] /; (n >= m (m + 1)/2) := T[n, m] = T[n - m, m - 1] + T[n - m, m]; T[_, _] = 0; a[n_] := T[n + 6, 3]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 30 2021 *) CROSSREFS Cf. A000070, A008951, A000097, A000710. Cf. A000041, A001399. Fourth column of Riordan triangle A008951 and of triangle A103923. Sequence in context: A325718 A011893 A132210 * A024827 A304792 A104161 Adjacent sequences: A000095 A000096 A000097 * A000099 A000100 A000101 KEYWORD nonn,easy AUTHOR N. J. A. Sloane EXTENSIONS Edited by Emeric Deutsch, Mar 23 2005 STATUS approved

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Last modified December 10 04:38 EST 2023. Contains 367699 sequences. (Running on oeis4.)