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A036043 Number of parts of partitions read by rows (in Abramowitz and Stegun order). 19
1, 1, 2, 1, 2, 3, 1, 2, 2, 3, 4, 1, 2, 2, 3, 3, 4, 5, 1, 2, 2, 2, 3, 3, 3, 4, 4, 5, 6, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6, 7, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 7, 8, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 8, 9 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The sequence of row lengths of this array is p(n) = A000041(n) (partition numbers).

The sequence of row sums is A006128(n), n>=1.

The number of times k appears in row n is A008284(n,k). - Franklin T. Adams-Watters, Jan 12 2006

The next level (row) gets created from each node by adding one or two more nodes. If a single node is added, its value is one more than the value of its parent. If two nodes are added, the first is equal in value to the parent and the value of the second is one more than the value of the parent. See A128628. - Alford Arnold, Mar 27 2007

REFERENCES

Abramowitz and Stegun, Handbook, p. 831, column labeled "m".

LINKS

T. D. Noe, Rows n = 1..25 of irregular triangle, flattened

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], p. 831.

Kevin Brown, Generalized Birthday Problem (N Items in M Bins), 1994-2010.

W. Lang: First 20 rows.

FORMULA

Sum_{k=1..n} k*A036043(n,n-k+1) = A066186(n). - L. Edson Jeffery, Aug 03 2013

EXAMPLE

1;

1, 2;

1, 2, 3;

1, 2, 2, 3, 4;

1, 2, 2, 3, 3, 4, 5;

1, 2, 2, 2, 3, 3, 3, 4, 4, 5, 6;

1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5,.

MAPLE

with(combinat): nmax:=9: for n from 1 to nmax do y(n):=numbpart(n): P(n):=sort(partition(n)): for k from 1 to y(n) do B(k) := P(n)[k] od: for k from 1 to y(n) do s:=0: j:=0: while s<n do j:=j+1: s := s + B(k)[j]: Q(n, k):=j; end do: od: od: seq(seq(Q(n, j),  j=1..y(n)), n=1..nmax); # [Johannes W. Meijer, Jun 21 2010, revised Nov 29 2012]

# alternative implementation based on A119441 by R. J. Mathar, Jul 12 2013

A036043 := proc(n, k)

    local pi;

    pi := ASPrts(n)[k] ;

    nops(pi) ;

end proc:

for n from 1 to 10 do

    for k from 1 to A000041(n) do

        printf("%d, ", A036043(n, k)) ;

    end do:

    printf("\n") ;

end do:

MATHEMATICA

Table[Map[Length, Partitions[n]], {n, 1, 8}] // Grid

CROSSREFS

Cf. A036036-A036040, A036042.

Cf. A049085, A080577.

Sequence in context: A094917 A082691 A183198 * A128628 A198338 A199086

Adjacent sequences:  A036040 A036041 A036042 * A036044 A036045 A036046

KEYWORD

nonn,easy,tabf

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 17 2001

STATUS

approved

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Last modified April 21 02:07 EDT 2014. Contains 240824 sequences.