%I A000070 M1054 N0396
%S A000070 1,2,4,7,12,19,30,45,67,97,139,195,272,373,508,684,915,1212,1597,2087,
2714,
%T A000070 3506,4508,5763,7338,9296,11732,14742,18460,23025,28629,35471,43820,53963,
%U A000070 66273,81156,99133,120770,146785,177970,215308,259891,313065,376326,451501
%N A000070 Sum_{k=0..n} p(k) where p(k) = number of partitions of k (A000041).
%C A000070 Number of partitions of n into parts but there are two kinds of parts
of size one.
%C A000070 Also number of graphical forest partitions of 2n+2.
%C A000070 a(n) = count 2 for each partition of n and 1 for each decrement. E.g.
the partitions of 4 are 4 (2), 31 (3), 22 (2), 211 (3) and 1111 (2).
2+3+2+3+2=12. This is related to the Ferrers representation. We can
see that taking the Ferrers diagram for each partition of n and adding
a new * to all available columns, we generate each partition of n+1,
but with repeats (A058884). - Jon Perry (perry(AT)globalnet.co.uk),
Feb 06 2004
%C A000070 Also the total number of all different integers in all partitions of
n. E.g. a(4)=7 because the partition of n=4 comprises the sets {1},
{1, 2},{2},{1, 3},{4} of different integers and their total number
is 7. - Thomas Wieder (wieder.thomas(AT)t-online.de), Apr 10 2004
%C A000070 Also the number of 1-transitions among all integer partitions of n. A
1-transition is the removal of a digit "1" from a partition containing
at least one "1" and subsequent addition of that "1" to another digit
in that partition. This other digit may be a "1" also, but all digits
of equal amount are considered as undistinquishable (unlabeled).
E.g. for n=6 one has the partition [1113] for which the following
two 1-transitions are possible: [1113] --> [123] and [1113] --> [114].
The 1-transitions of n form a partial order (poset). For n=6 one
has 12 1-transitions: [111111] --> [11112], [11112] --> [1113], [11112]
--> [1122], [1113] --> [114], [1113] --> [123], [1122] --> [123],
[1122] --> [222], [123] --> [33], [123] --> [24], [114] --> [15],
[114] --> [24], [15] --> [6]. - Thomas Wieder (wieder.thomas(AT)t-online.de),
Mar 08 2005
%C A000070 With offset 1, also the number of 1's in all partitions of n. For example,
3 = 2+1 = 1+1+1, a(3) = (zero 1's) + (one 1's) + (three 1's), so
a(3) = 4. - Naohiro Nomoto (n_nomoto(AT)yabumi.com), Jan 09 2002.
See the Riordan reference p. 184, last formula, first term, for a
proof based on Fine's identity given in Riordan, p. 182 (20).
%C A000070 Also number of partitions of 2n+1 where one of the parts is greater than
n (also where there are more than n parts) and of 2n+2 where one
of the parts is greater than n+1 (or with more than n+1 parts). -
Henry Bottomley (se16(AT)btinternet.com), Aug 01 2005
%C A000070 Equals left border of triangle A137633 - Gary W. Adamson (qntmpkt(AT)yahoo.com),
Jan 31 2008
%C A000070 Equals row sums of triangle A027293 [From Gary W. Adamson (qntmpkt(AT)yahoo.com),
Oct 26 2008]
%C A000070 Convolved with A010815 = [1,1,1,...]. n-th partial sum of A000041 convolved
with A010815 = the binomial sequence starting (1, n,...). [From Gary
W. Adamson (qntmpkt(AT)yahoo.com), Nov 09 2008]
%C A000070 Equals A036469 convolved with A035363. [From Gary W. Adamson (qntmpkt(AT)yahoo.com),
Jun 09 2009]
%D A000070 C. C. Cadogan, On partly ordered partitions of a positive integer, Fib.
Quart., 9 (1971), 329-336.
%D A000070 H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables,
Vol. 4, Cambridge Univ. Press, 1958, p. 90.
%D A000070 R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 6.
%D A000070 A. M. Odlyzko, Asymptotic Enumeration Methods, p. 19
%D A000070 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.
%D A000070 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A000070 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%H A000070 T. D. Noe, <a href="b000070.txt">Table of n, a(n) for n=0..1000</a>
%H A000070 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 A000070 P. Flajolet and B. Salvy, <a href="http://www.expmath.org/expmath/volumes/
7/7.html">Euler sums and contour integral representations</a>, Experimental
Mathematics, Vol. 7 Issue 1 (1998)
%H A000070 D. Frank, C. D. Savage and J. A. Sellers, <a href="http://bkocay.cs.umanitoba.ca/
ArsComb/ArsComb.html">On the Number of Graphical Forest Partitions</
a>, Ars Combinatoria, Vol. 65 (2002), 33-37.
%H A000070 H. Fripertinger, <a href="http://www-ang.kfunigraz.ac.at/~fripert/fga/
k1partn.html">Partitions of an Integer</a>
%H A000070 N. Hobson, Nick's Mathematical Puzzles, <a href="http://www.qbyte.org/
puzzles/p056s.html">Partition identity (or a proof of Stanley's Theorem)</
a>
%H A000070 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=113">
Encyclopedia of Combinatorial Structures 113</a>
%H A000070 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=126">
Encyclopedia of Combinatorial Structures 126</a>
%H A000070 G. Pfeiffer, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Counting
Transitive Relations</a>, Journal of Integer Sequences, Vol. 7 (2004),
Article 04.3.2.
%H A000070 F. Ruskey, <a href="http://www.theory.cs.uvic.ca/~cos/inf/nump/NumPartition.html">
Combinatorial Objects Server</a>
%H A000070 N. J. A. Sloane, <a href="transforms.txt">Transforms</a>
%H A000070 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
StanleysTheorem.html">Link to a section of The World of Mathematics.</
a>
%F A000070 Euler transform of 2 1 1 1 1 1 1...
%F A000070 log(a(n)) ~ -3.3959 + 2.44613*Sqrt(n). - Robert G. Wilson v (rgwv(AT)rgwv.com),
Jan 11 2002
%F A000070 a(n)=1/n*Sum_{k=1..n} (sigma(k)+1)*a(n-k), n > 1, a(0)=1. - Vladeta Jovovic
(vladeta(AT)eunet.rs), Aug 22 2002
%F A000070 G.f.: (1/(1-x))*Product(1/(1-x^m)), m=1..inf.
%F A000070 Sequence seems to have the same parity as A027349. Comment from James
Sellers, Mar 08 2006: that is true.
%F A000070 a(n) =(1+O(n^(-1/6)))/(2*C*sqrt(3n))*exp(C*sqrt(n)) where C=Pi*sqrt(2/
3) - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 31 2004
%F A000070 a(n) =A000041(2n+1)-A110618(2n+1) =A000041(2n+2)-A110618(2n+2) - Henry
Bottomley (se16(AT)btinternet.com), Aug 01 2005
%F A000070 Row sums of triangle A133735 - Gary W. Adamson (qntmpkt(AT)yahoo.com),
Sep 22 2007
%p A000070 a:=n->(sum((numbpart(j)), j=0..n)):seq(a(n), n=0..44);# [From Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), Aug 26 2008]
%t A000070 CoefficientList[ Series[1/(1 - x)*Product[1/(1 - x^k), {k, 75}], {x,
0, 45}], x] (from Robert G. Wilson v Jul 13 2004)
%t A000070 (* First do *) Needs["Combinatorica`"] (* then *) Table[ Count[ Flatten@
Partitions@ n, 1], {n, 45}] (from Robert G. Wilson v, (rgwv(AT)rgwv.com)
Aug 06 2008)
%o A000070 (PARI) a(n)=if(n<0,0,polcoeff(1/prod(m=1,n,1-x^m,1+x*O(x^n))/(1-x),n))
%Y A000070 Row sums of triangle A092905.
%Y A000070 Cf. A014153, A024786, A026905, A058884, A093694, A133735, A137633.
%Y A000070 A027293 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 26 2008]
%Y A000070 A010815 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 09 2008]
%Y A000070 A036469, A035363 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 09
2009]
%Y A000070 Sequence in context: A079719 A036439 A035298 this_sequence A008609 A100823
A102346
%Y A000070 Adjacent sequences: A000067 A000068 A000069 this_sequence A000071 A000072
A000073
%K A000070 nonn,easy,nice
%O A000070 0,2
%A A000070 N. J. A. Sloane (njas(AT)research.att.com).
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