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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) 14
 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 Fausto A. C. Cariboni, Table of n, a(n) for n = 1..140 (terms 1..89 from Alois P. Heinz) N. Metropolis and P. R. Stein, An elementary solution to a problem in restricted partitions, J. Combin. Theory, 9 (1970), 365-376. Vladimir A. Shlyk, Number of Vertices of the Polytope of Integer Partitions and Factorization of the Partitioned Number, arXiv:1805.07989 [math.CO], 2018. 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`(xn, 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-François 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: A285460 A236429 A316245 * A006906 A324703 A120940 Adjacent sequences:  A002216 A002217 A002218 * A002220 A002221 A002222 KEYWORD nonn,nice AUTHOR 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 October 2 23:51 EDT 2022. Contains 357230 sequences. (Running on oeis4.)