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A000097 Number of partitions of n if there are two kinds of 1's and two kinds of 2's.
(Formerly M1361 N0525)
21
1, 2, 5, 9, 17, 28, 47, 73, 114, 170, 253, 365, 525, 738, 1033, 1422, 1948, 2634, 3545, 4721, 6259, 8227, 10767, 13990, 18105, 23286, 29837, 38028, 48297, 61053, 76926, 96524, 120746, 150487, 187019, 231643, 286152, 352413, 432937, 530383, 648245 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Also number of partitions of 2*n with exactly 2 odd parts (offset 1). - Vladeta Jovovic, Jan 12 2005

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

From Christian Gutschwager (gutschwager(AT)math.uni-hannover.de), Feb 10 2010: (Start)

a(n) is also the number of pairs of partitions of n+2 which differ by only one box (for bijection see Gutschwager link).

a(n) is also the number of partitions of n with two parts marked.

a(n) is also the number of partitions of n+1 with two different parts marked. (End)

Convolution of A000041 and A008619. - Vaclav Kotesovec, Aug 18 2015

A000097(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

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

REFERENCES

H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 90.

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.

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).

LINKS

T. D. Noe and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

Christian Gutschwager, Skew characters which contain only few components, arXiv:1002.1610 [math.CO], 2010-2011.

C. Gutschwager, Reduced Kronecker products which are multiplicity free or contain only a few components, Eur. J. Combinat. 31 (2010) 1996-2005 doi:10.1016/j.ejc.2010.05.008.

N. J. A. Sloane, Transforms

FORMULA

Euler transform of 2 2 1 1 1 1 1...

G.f.: 1/( (1-x) * (1-x^2) * prod(k>=1, 1-x^k) ).

a(n) = sum(A000070(n-2*j), j=0..floor(n/2)), n>=0.

a(n) = A014153(n)/2 + A087787(n)/4 + A000070(n)/4. - Vaclav Kotesovec, Nov 05 2016

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

EXAMPLE

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'.

MAPLE

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

MATHEMATICA

CoefficientList[Series[1/((1 - x) (1 - x^2) Product[1 - x^k, {k, 1, 100}]), {x, 0, 100}], x] (* Ben Branman, Mar 07 2012 *)

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-Fran├žois Alcover, Apr 09 2014, after Alois P. Heinz *)

(1/((1 - x) (1 - x^2) QPochhammer[x]) + O[x]^50)[[3]] (* Vladimir Reshetnikov, Nov 22 2016 *)

PROG

(PARI) x = 'x + O('x^66); Vec( 1/((1-x)*(1-x^2)*eta(x)) ) \\ Joerg Arndt, Apr 29 2013

CROSSREFS

First differences are in A024786.

Cf. A000070, A008951, A000098, A000710.

Third column of Riordan triangle A008951 and of triangle A103923.

Sequence in context: A139672 A093694 A068006 * A081996 A034329 A230441

Adjacent sequences:  A000094 A000095 A000096 * A000098 A000099 A000100

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 04 2004

Edited by Emeric Deutsch, Mar 23 2005

More terms from Franklin T. Adams-Watters, Mar 20 2006

Edited by Charles R Greathouse IV, Apr 20 2010

STATUS

approved

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Last modified January 16 21:37 EST 2019. Contains 319206 sequences. (Running on oeis4.)