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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
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
Fourth column of Riordan triangle A008951 and of triangle A103923.
Sequence in context: A325718 A011893 A132210 * A024827 A304792 A104161
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Edited by Emeric Deutsch, Mar 23 2005
STATUS
approved

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Last modified March 19 01:34 EDT 2024. Contains 370952 sequences. (Running on oeis4.)