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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)
10
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

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

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

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

CROSSREFS

Cf. A000070, A008951, A000097, A000710.

Fourth column of Riordan triangle A008951 and of triangle A103923.

Sequence in context: A018739 A011893 A132210 * A024827 A104161 A288579

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 November 22 09:35 EST 2017. Contains 295076 sequences.