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.

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.yu), 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

njas

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

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