

A036043


Irregular triangle read by rows: row n (n >= 0) gives number of parts in all partitions of n (in Abramowitz and Stegun order).


21



0, 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
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OFFSET

0,4


COMMENTS

The sequence of row lengths of this array is p(n) = A000041(n) (partition numbers).
The sequence of row sums is A006128(n).
The number of times k appears in row n is A008284(n,k).  Franklin T. AdamsWatters, 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
The 1's in the (flattened) sequence mark the start of a new row, the value that precedes the 1 equals the row number minus one. (I.e., the 1 preceded by a 0 is the start of row 1, the 1 preceded by a 6 is the start of row 7, etc.)  M. F. Hasler, Jun 06 2018


REFERENCES

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


LINKS

T. D. Noe, Rows n = 0..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), 19942010.
Wolfdieter Lang, Rows n = 1 ..20.


EXAMPLE

0;
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;


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: 0, 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:


PROG

(PARI) A036043(n, k)=#partitions(n)[k] \\ M. F. Hasler, Jun 06 2018
(SageMath) from collections import Counter
def A036043_row(n):
return [len(p) for k in (0..n) for p in Partitions(n, length=k)]
for n in (0..10): print(A036043_row(n)) # Peter Luschny, Nov 02 2019


CROSSREFS

Cf. A036036, A036037, A036038, A036039, A036040, A036042.
Cf. A049085, A080577.
Sequence in context: A249160 A269970 A252230 * A238966 A128628 A275723
Adjacent sequences: A036040 A036041 A036042 * A036044 A036045 A036046


KEYWORD

nonn,easy,tabf,changed


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 17 2001
a(0) inserted by Franklin T. AdamsWatters, Jun 24 2014
Incorrect formula deleted by M. F. Hasler, Jun 06 2018


STATUS

approved



