Search: id:A001970
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%I A001970 M2576 N1019
%S A001970 1,1,3,6,14,27,58,111,223,424,817,1527,2870,5279,9710,17622,
%T A001970 31877,57100,101887,180406,318106,557453,972796,1688797,2920123,
%U A001970 5026410,8619551,14722230,25057499,42494975,71832114,121024876
%N A001970 Functional determinants; partitions of partitions; Euler transform applied 
               twice to all 1's sequence.
%C A001970 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.
%C A001970 Equivalently (Cayley), a(n) = number of 2-dimensional partitions of n. 
               E.g. for n = 4 we have:
%C A001970 4.31.3.22.2.211.21.2..2.1111.111.11.11.1
%C A001970 .....1....2.....1..11.1......1...11.1..1
%C A001970 ......................1.............1..1
%C A001970 .......................................1
%C A001970 Also total number of different species of singularity for conjugate functions 
               with n letters (Sylvester).
%D A001970 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.
%D A001970 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.
%D A001970 R. Kaneiwa. An asymptotic formula for Cayley's double partition function 
               p(2; n). Tokyo J. Math. 2, 137-158 (1979).
%D A001970 V. A. Liskovets, Counting rooted initially connected directed graphs. 
               Vesci Akad. Nauk. BSSR, ser. fiz.-mat., No 5, 23-32 (1969), MR44 
               #3927.
%D A001970 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001970 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001970 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.
%D A001970 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.
%H A001970 T. D. Noe, <a href="b001970.txt">Table of n, a(n) for n=1..500</a>
%H A001970 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A001970 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=148">
               Encyclopedia of Combinatorial Structures 148</a>
%H A001970 N. J. A. Sloane, <a href="transforms.txt">Transforms</a>
%H A001970 N. J. A. Sloane and Thomas Wieder, <a href="http://arXiv.org/abs/math.CO/
               0307064">The Number of Hierarchical Orderings</a>, Order 21 (2004), 
               83-89.
%H A001970 J. J. Sylvester, The collected mathematical papers of James Joseph Sylvester, 
               <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;
               c=umhistmath;idno=AAS8085.0002.001">vol. 2</a>, <a href="http://www.hti.umich.edu/
               cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;c=umhistmath;
               idno=AAS8085.0003.001">vol. 3</a>, <a href="http://www.hti.umich.edu/
               cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;c=umhistmath;
               idno=AAS8085.0004.001">vol. 4</a>.
%H A001970 <a href="Sindx_Ro.html#rooted">Index entries for sequences related to 
               rooted trees</a>
%F A001970 G.f.: Product_{k >= 1} 1/(1-x^k)^p(k), where p(k) = number of partitions 
               of k = A000041. [Cayley]
%F A001970 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
%e A001970 a(3) = 6 because we have (111) = (111) = (11)(1) = (1)(1)(1), (12) = 
               (12) = (1)(2), (3) = (3)
%p A001970 with(combstruct); SetSetSetU := [T, {T=Set(S), S=Set(U,card >= 1), U=Set(Z,
               card >=1)},unlabeled];
%Y A001970 Cf. A000041, A061259, A006171, A061255, A061256, A061257, A089292, A000219.
%Y A001970 Cf. A089300.
%Y A001970 Related to A001383 via generating function.
%Y A001970 Sequence in context: A049940 A051749 A030012 this_sequence A006951 A132891 
               A055890
%Y A001970 Adjacent sequences: A001967 A001968 A001969 this_sequence A001971 A001972 
               A001973
%K A001970 nonn,nice,easy
%O A001970 1,3
%A A001970 N. J. A. Sloane (njas(AT)research.att.com).
%E A001970 Additional comments from Valery A.Liskovets (liskov(AT)im.bas-net.by)
%E A001970 Sylvester references from Barry Cipra (bcipra(AT)rconnect.com), Oct 07 
               2003

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