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A225751 Number of different figures obtained by a putting two Young diagrams of partitions lambda and mu, such that |lambda| + |mu| = n on top of each other. 1
1, 1, 3, 5, 10, 15, 26, 38, 60, 85, 127, 176, 253, 343, 478, 639, 870, 1145, 1530, 1990, 2617, 3367, 4369, 5568, 7143, 9024, 11460, 14369, 18087, 22517, 28121, 34787, 43136, 53048, 65358, 79944, 97921, 119173, 145188, 175883, 213221, 257177, 310351, 372820 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

See Mukhin reference item 4.3. 'A challenge', sum of p(k) for k = ceiling(n/2) to n with p(k) the partition numbers A000041. Remark that the indexing in the reference misses N_3 (should be N_3=5 and so on).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Eugene Mukhin, Symmetric polynomials and partitions

FORMULA

a(n) = Sum_{i=ceiling(n/2)..n} A000041(i).

a(n) = A000070(n) - A000070(ceiling(n/2)-1). - Alois P. Heinz, Jul 31 2016

EXAMPLE

a(3) = 5 is illustrated by the following 5 different results:

{2}   = {1}  &  {2}

{2}   = {2}  &  {1}

{3}   = { }  &  {3}

{1,1} = {1}  &  {1,1}

{1,1} = {1,1}&  {1}

{2,1} =  { } &  {2,1}

{1,1,1}= { } &  {1,1,1}

producing {2}, {3}, {1,1}, {2,1} and {1,1,1} as superpositions of two partitions with sum of lengths = 3.

MAPLE

with(combinat):

a:= n-> add(numbpart(i), i=ceil(n/2)..n):

seq(a(n), n=0..50);  # Alois P. Heinz, May 15 2013

MATHEMATICA

Table[Sum[PartitionsP[k], {k, Ceiling[n/2], n}], {n, 36}]

PROG

(PARI) a(n)=sum(k=ceil(n/2), n, numbpart(k)); \\ Joerg Arndt, May 15 2013

CROSSREFS

Cf. A000041, A000070.

Sequence in context: A090491 A126728 A070557 * A264397 A254346 A132302

Adjacent sequences:  A225748 A225749 A225750 * A225752 A225753 A225754

KEYWORD

nonn

AUTHOR

Wouter Meeussen, May 14 2013

STATUS

approved

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Last modified August 22 02:44 EDT 2019. Contains 326169 sequences. (Running on oeis4.)