1,3

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).

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).

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.

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

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 (vladeta(AT)eunet.rs), Apr 21 2001

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

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

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

Cf. A089300.

Related to A001383 via generating function.

Sequence in context: A049940 A051749 A030012 this_sequence A006951 A132891 A055890

Adjacent sequences: A001967 A001968 A001969 this_sequence A001971 A001972 A001973

nonn,nice,easy

N. J. A. Sloane (njas(AT)research.att.com).

Additional comments from Valery A.Liskovets (liskov(AT)im.bas-net.by)

Sylvester references from Barry Cipra (bcipra(AT)rconnect.com), Oct 07 2003