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A002219 a(n) is the number of partitions of 2n that can be obtained by adding together two (not necessarily distinct) partitions of n.
(Formerly M2574 N1018)
8
1, 3, 6, 14, 25, 53, 89, 167, 278, 480, 760, 1273, 1948, 3089, 4682, 7177, 10565, 15869, 22911, 33601, 47942, 68756, 96570, 136883, 189674, 264297, 362995, 499617, 678245, 924522, 1243098, 1676339, 2237625, 2988351, 3957525, 5247500, 6895946, 9070144, 11850304 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

REFERENCES

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

Alois P. Heinz, Table of n, a(n) for n = 1..89

N. Metropolis and P. R. Stein, An elementary solution to a problem in restricted partitions, J. Combin. Theory, 9 (1970), 365-376.

FORMULA

See A213074 for Metropolis and Stein's formulas.

a(n) = A000041(2*n) - A006827(n) = A000041(2*n) - A046663(2*n,n)

EXAMPLE

Here are the seven partitions of 5: 1^5, 1^3 2, 1 2^2, 1^2 3, 2 3, 1 4, 5. Adding these together in pairs we get a(5) = 25 partitions of 10: 1^10, 1^8 2, 1^6 2^2, etc. (we get all partitions of 10 into parts of size <= 5 - there are 30 such partitions - except for five of them: we do not get 2 4^2, 3^2 4, 2^3 4, 1 3^3, 2^5). N. J. A. Sloane, Jun 03 2012

MAPLE

g:= proc(n, i) option remember;

     `if`(n=0, 1, `if`(i>1, g(n, i-1), 0)+`if`(i>n, 0, g(n-i, i)))

    end:

b:= proc(n, i, s) option remember;

     `if`(i=1 and s<>{} or n in s, g(n, i), `if`(i<1 or s={}, 0,

      b(n, i-1, s)+ `if`(i>n, 0, b(n-i, i, map(x-> {`if`(x>n-i, NULL,

      max(x, n-i-x)), `if`(x<i or x>n, NULL, max(x-i, n-x))}[], s)))))

    end:

a:= n-> b(2*n, n, {n}):

seq(a(n), n=1..25);  # Alois P. Heinz, Jul 10 2012

MATHEMATICA

b[n_, i_, s_] := b[n, i, s] = If[MemberQ[s, 0 | n], 0, If[n == 0, 1, If[i < 1, 0, b[n, i-1, s] + If[i <= n, b[n-i, i, Select[Flatten[Transpose[{s, s-i}]], 0 <= # <= n-i &]], 0]]]]; A006827[n_] := b[2*n, 2*n, {n}]; a[n_] := PartitionsP[2*n] - A006827[n]; Table[Print[an = a[n]]; an, {n, 1, 25}] (* Jean-Fran├žois Alcover, Nov 12 2013, after Alois P. Heinz *)

CROSSREFS

Cf. A064914, A000041, A002220, A002221, A002222, A213074, A006827, A046663.

Column m=2 of A213086.

Sequence in context: A291988 A285460 A236429 * A006906 A120940 A049940

Adjacent sequences:  A002216 A002217 A002218 * A002220 A002221 A002222

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Better description from Vladeta Jovovic, Mar 06 2000

More terms from Christian G. Bower, Oct 12 2001

Edited by N. J. A. Sloane, Jun 03 2012

More terms from Alois P. Heinz, Jul 10 2012

STATUS

approved

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Last modified November 23 20:23 EST 2017. Contains 295141 sequences.