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%I M1361 N0525 #151 Oct 13 2023 10:01:57
%S 1,2,5,9,17,28,47,73,114,170,253,365,525,738,1033,1422,1948,2634,3545,
%T 4721,6259,8227,10767,13990,18105,23286,29837,38028,48297,61053,76926,
%U 96524,120746,150487,187019,231643,286152,352413,432937,530383,648245
%N Number of partitions of n if there are two kinds of 1's and two kinds of 2's.
%C Also number of partitions of 2*n with exactly 2 odd parts (offset 1). - _Vladeta Jovovic_, Jan 12 2005
%C Also number of transitions from one partition of n+2 to another, where a transition consists of replacing any two parts with their sum. Remove all 1' and 2' from the partition, replacing them with ((number of 2') + 1) and ((number of 1') + (number of 2') + 1); these are the two parts being summed. Number of partitions of n into parts of 2 kinds with at most 2 parts of the second kind, or of n+2 into parts of 2 kinds with exactly 2 parts of the second kind. - _Franklin T. Adams-Watters_, Mar 20 2006
%C From Christian Gutschwager (gutschwager(AT)math.uni-hannover.de), Feb 10 2010: (Start)
%C a(n) is also the number of pairs of partitions of n+2 which differ by only one box (for bijection see Gutschwager link).
%C a(n) is also the number of partitions of n with two parts marked.
%C a(n) is also the number of partitions of n+1 with two different parts marked. (End)
%C Convolution of A000041 and A008619. - _Vaclav Kotesovec_, Aug 18 2015
%C a(n) = P(/2,n), a particular case of P(/k,n) defined as follows: P(/0,n) = A000041(n) and P(/k,n) = P(/k-1, n) + P(/k-1,n-k) + P(/k-1, n-2k) + ... Also, P(/k,n) = the convolution of A000041 and the partitions of n with exactly k parts, and g.f. P(/k,n) = (g.f. for P(n)) * 1/(1-x)...(1-x^k). - _Gregory L. Simay_, Mar 22 2018
%C a(n) is also the sum of binomial(D(p),2) in partitions p of (n+3), where D(p)= number of different sizes of parts in p. - _Emily Anible_, Apr 03 2018
%C Also partitions of 2*(n+1) with alternating sum 2. Also partitions of 2*(n+1) with reverse-alternating sum -2 or 2. - _Gus Wiseman_, Jun 21 2021
%C Define the distance graph of the partitions of n using the distance function in A366156 as follows: two vertices (partitions) share an edge if and only if the distance between the vertices is 2. Then a(n) is the number of edges in the distance graph of the partitions of n. - _Clark Kimberling_, Oct 12 2023
%D H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 90.
%D J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe and Vaclav Kotesovec, <a href="/A000097/b000097.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)
%H P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H Christian Gutschwager, <a href="http://arxiv.org/abs/1002.1610">Skew characters which contain only few components</a>, arXiv:1002.1610 [math.CO], 2010-2011.
%H Christian Gutschwager, <a href="https://dx.doi.org/10.1016/j.ejc.2010.05.008">Reduced Kronecker products which are multiplicity free or contain only a few components</a>, Eur. J. Combinat. 31 (2010) 1996-2005. doi:10.1016/j.ejc.2010.05.008.
%H J. P. Robinson, <a href="https://doi.org/10.1016/0097-3165(88)90009-X">Edges in the poset of partitions of an integer</a>, J. Combin. Theory Ser. A, 48 (1988), 236-238.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F Euler transform of 2 2 1 1 1 1 1...
%F G.f.: 1/( (1-x) * (1-x^2) * Product_{k>=1} (1-x^k) ).
%F a(n) = Sum_{j=0..floor(n/2)} A000070(n-2*j), n>=0.
%F a(n) = A014153(n)/2 + A087787(n)/4 + A000070(n)/4. - _Vaclav Kotesovec_, Nov 05 2016
%F a(n) ~ sqrt(3) * exp(Pi*sqrt(2*n/3)) / (4*Pi^2) * (1 + 35*Pi/(24*sqrt(6*n))). - _Vaclav Kotesovec_, Aug 18 2015, extended Nov 05 2016
%F a(n) = A120452(n) + A344741(n). - _Gus Wiseman_, Jun 21 2021
%e a(3) = 9 because we have 3, 2+1, 2+1', 2'+1, 2'+1', 1+1+1, 1+1+1', 1+1'+1' and 1'+1'+1'.
%e From _Gus Wiseman_, Jun 22 2021: (Start)
%e The a(0) = 1 through a(4) = 9 partitions of 2*(n+1) with exactly 2 odd parts:
%e (1,1) (3,1) (3,3) (5,3)
%e (2,1,1) (5,1) (7,1)
%e (3,2,1) (3,3,2)
%e (4,1,1) (4,3,1)
%e (2,2,1,1) (5,2,1)
%e (6,1,1)
%e (3,2,2,1)
%e (4,2,1,1)
%e (2,2,2,1,1)
%e The a(0) = 1 through a(4) = 9 partitions of 2*(n+1) with alternating sum 2:
%e (2) (3,1) (4,2) (5,3)
%e (2,1,1) (2,2,2) (3,3,2)
%e (3,2,1) (4,3,1)
%e (3,1,1,1) (3,2,2,1)
%e (2,1,1,1,1) (4,2,1,1)
%e (2,2,2,1,1)
%e (3,2,1,1,1)
%e (3,1,1,1,1,1)
%e (2,1,1,1,1,1,1)
%e (End)
%p with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n->`if`(n<3,2,1)): seq(a(n), n=0..40); # _Alois P. Heinz_, Sep 08 2008
%t CoefficientList[Series[1/((1 - x) (1 - x^2) Product[1 - x^k, {k, 1, 100}]), {x, 0, 100}], x] (* _Ben Branman_, Mar 07 2012 *)
%t etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b]; a = etr[If[# < 3, 2, 1]&]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Apr 09 2014, after _Alois P. Heinz_ *)
%t (1/((1 - x) (1 - x^2) QPochhammer[x]) + O[x]^50)[[3]] (* _Vladimir Reshetnikov_, Nov 22 2016 *)
%t Table[Length@IntegerPartitions[n,All,Join[{1,2},Range[n]]],{n,0,15}] (* _Robert Price_, Jul 28 2020 and Jun 21 2021 *)
%t T[n_, 0] := PartitionsP[n];
%t T[n_, m_] /; (n >= m (m + 1)/2) := T[n, m] = T[n - m, m - 1] + T[n - m, m];
%t T[_, _] = 0;
%t a[n_] := T[n + 3, 2];
%t Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, May 30 2021 *)
%t ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];Table[Length[Select[IntegerPartitions[n],ats[#]==2&]],{n,0,30,2}] (* _Gus Wiseman_, Jun 21 2021*)
%o (PARI) my(x = 'x + O('x^66)); Vec( 1/((1-x)*(1-x^2)*eta(x)) ) \\ _Joerg Arndt_, Apr 29 2013
%Y First differences are in A024786.
%Y Cf. A000098, A000710.
%Y Third column of Riordan triangle A008951 and of triangle A103923.
%Y The case of reverse-alternating sum 1 or alternating sum 0 is A000041.
%Y The case of reverse-alternating sum -1 or alternating sum 1 is A000070.
%Y The normal case appears to be A004526 or A065033.
%Y The strict case is A096914.
%Y The case of reverse-alternating sum 2 is A120452.
%Y The case of reverse-alternating sum -2 is A344741.
%Y A001700 counts compositions with alternating sum 2.
%Y A035363 counts partitions into even parts.
%Y A058696 counts partitions of 2n.
%Y A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A124754 gives alternating sums of standard compositions (reverse: A344618).
%Y A316524 is the alternating sum of the prime indices of n (reverse: A344616).
%Y A344610 counts partitions by sum and positive reverse-alternating sum.
%Y A344611 counts partitions of 2n with reverse-alternating sum >= 0.
%Y Cf. A006330, A027187, A239830, A306145, A343941, A344607, A344608, A344619, A344650, A344651, A344740.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E More terms from Pab Ter (pabrlos(AT)yahoo.com), May 04 2004
%E Edited by _Emeric Deutsch_, Mar 23 2005
%E More terms from _Franklin T. Adams-Watters_, Mar 20 2006
%E Edited by _Charles R Greathouse IV_, Apr 20 2010
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