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A001970 Functional determinants; partitions of partitions; Euler transform applied twice to all 1's sequence.
(Formerly M2576 N1019)
19
1, 1, 3, 6, 14, 27, 58, 111, 223, 424, 817, 1527, 2870, 5279, 9710, 17622, 31877, 57100, 101887, 180406, 318106, 557453, 972796, 1688797, 2920123, 5026410, 8619551, 14722230, 25057499, 42494975, 71832114, 121024876 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

a(n) = number of partitions of n, when for each k there are p(k) different copies of part k. E.g., let the parts be 1, 2a, 2b, 3a, 3b, 3c, 4a, 4b, 4c, 4d, 4e, ... Then the a(4) = 14 partitions of 4 are: 4 = 4a = 4b = ... = 4e = 3a+1 = 3b+1 = 3c+1 = 2a+2a = 2a+2b = 2b+2b = 2a+1 = 2b+1 = 1+1+1+1.

Equivalently (Cayley), a(n) = number of 2-dimensional partitions of n. E.g. for n = 4 we have:

4.31.3.22.2.211.21.2..2.1111.111.11.11.1

.....1....2.....1..11.1......1...11.1..1

......................1.............1..1

.......................................1

Also total number of different species of singularity for conjugate functions with n letters (Sylvester).

REFERENCES

P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.

A. Cayley, Recherches sur les matrices dont les termes sont des fonctions line'aires d'une seule inde'termine'e, J. Reine angew. Math., 50 (1855), 313-317; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, p. 219.

R. Kaneiwa. An asymptotic formula for Cayley's double partition function p(2; n). Tokyo J. Math. 2, 137-158 (1979).

L. Kaylor and D. Offner, Counting Matrices Over a Finite Field With All Eigenvalues in the Field, 2013; http://www.westminster.edu/staff/offnerde/CountingEigenvalues.pdf

XiKun Li, JunLi Li, Bin Liu and CongFeng Qiao, The parametric symmetry and numbers of the entangled class of 2 × M × N system, SCIENCE CHINA PHYSICS, MECHANICS & ASTRONOMY, Volume 54, Number 8, 1471-1475, DOI: 10.1007/s11433-011-4395-9; http://www.springerlink.com/content/24g51u87527017u7/

V. A. Liskovets, Counting rooted initially connected directed graphs. Vesci Akad. Nauk. BSSR, ser. fiz.-mat., No 5, 23-32 (1969), MR44 #3927.

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).

J. J. Sylvester, An Enumeration of the Contacts of Lines and Surfaces of the Second Order, Phil. Mag. 1 (1851), 119-140. Reprinted in Collected Papers, Vol. 1. See p. 239, where one finds a(n)-2, but with errors.

J. J. Sylvester, Note on the 'Enumeration of the Contacts of Lines and Surfaces of the Second Order, Phil. Mag., Vol. VII (1854), pp. 331-334. Reprinted in Collected Papers, Vol. 2, pp. 30-33.

LINKS

T. D. Noe, Table of n, a(n) for n=1..500

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 148

N. J. A. Sloane, Transforms

N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.

J. J. Sylvester, The collected mathematical papers of James Joseph Sylvester, vol. 2, vol. 3, vol. 4.

Index entries for sequences related to rooted trees

FORMULA

G.f.: Product_{k >= 1} 1/(1-x^k)^p(k), where p(k) = number of partitions of k = A000041. [Cayley]

a(n) = (1/n)*Sum_{k = 1..n} a(n-k)*b(k), n>1, a(0) = 1, b(k) = Sum_{d|k} d*numbpart(d), where numbpart(d) = number of partitions of d, cf. A061259. - Vladeta Jovovic, Apr 21 2001

Logarithmic derivative yields A061259 (equivalent to above formula from Vladeta Jovovic). - Paul D. Hanna, Sep 05 2012

EXAMPLE

a(3) = 6 because we have (111) = (111) = (11)(1) = (1)(1)(1), (12) = (12) = (1)(2), (3) = (3).

MAPLE

with(combstruct); SetSetSetU := [T, {T=Set(S), S=Set(U, card >= 1), U=Set(Z, card >=1)}, unlabeled];

MATHEMATICA

m = 32; f[x_] = Product[1/(1-x^k)^PartitionsP[k], {k, 1, m}]; CoefficientList[ Series[f[x], {x, 0, m-1}], x] (* From Jean-François Alcover, Jul 19 2011, after g.f. *)

CROSSREFS

Cf. A000041, A061259, A006171, A061255, A061256, A061257, A089292, A000219.

Cf. A089300.

Related to A001383 via generating function.

Sequence in context: A049940 A051749 A030012 * A006951 A224840 A132891

Adjacent sequences:  A001967 A001968 A001969 * A001971 A001972 A001973

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Additional comments from Valery A. Liskovets

Sylvester references from Barry Cipra, Oct 07 2003

STATUS

approved

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Last modified April 18 19:30 EDT 2014. Contains 240733 sequences.