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%I #26 Feb 15 2022 13:03:35
%S 1,1,1,1,1,2,1,1,2,3,1,1,2,3,4,1,1,2,3,4,5,1,1,2,3,4,5,6,1,1,2,3,4,5,
%T 6,7,1,1,2,3,4,5,6,7,8,1,1,2,3,4,5,6,7,8,9,1,1,2,3,4,5,6,7,8,9,10,1,1,
%U 2,3,4,5,6,7,8,9,10,11
%N A002260 preceded by a column of 1's: a (1, 1, 2, 3, 4, 5, ...) crescendo triangle by rows.
%C Row sums = A000124.
%C Eigensequence of the triangle = A000142, the factorials.
%C The triangle as an infinite lower triangular matrix * [1,2,3,...] = A064999.
%C Generated from A128227 by rotating each row by one position to the right. - _R. J. Mathar_, Sep 25 2008
%C A sequence B is called a reluctant sequence of sequence A, if B is triangle array read by rows: row number k coincides with first k elements of the sequence A. Sequence A144328 is the reluctant sequence of A028310 (1 followed by the natural numbers). - _Boris Putievskiy_, Dec 12 2012
%C If offset were changed to 0, a(n) would equal the
%C Let S_n be the set of partitions of n into distinct parts where the number of parts is maximal for that n. For example, for n=6, the set S_6 consists of just one such partition: S_6={1,2,3}. Similarly, for n=7, S_7={1,2,4}, But for n=8, S_8 will contain two partitions S_8= { {1,2,5}, {1,3,4} }. Then |S(n)| = a(n+1). Cf. A178702. - _David S. Newman_ and _Benoit Jubin_, Dec 13 2010
%H Reinhard Zumkeller, <a href="/A144328/b144328.txt">Rows n = 1..100 of triangle, flattened</a>
%H Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a>, arXiv:1212.2732 [math.CO], 2012.
%F Triangle A002260 (natural numbers crescendo triangle) preceded by a column of 1's, = a (1, 1, 2, 3, 4, 5, ...) crescendo triangle by rows.
%F a(n) = A028310(m), where m = n-t(t+1)/2, t = floor((-1+sqrt(8*n-7))/2)). - _Boris Putievskiy_, Dec 13 2012
%e First few rows of the triangle:
%e 1;
%e 1, 1;
%e 1, 1, 2;
%e 1, 1, 2, 3;
%e 1, 1, 2, 3, 4;
%e 1, 1, 2, 3, 4, 5;
%e ...
%t Flatten[Table[Join[{1},Range[n]],{n,0,11}]] (* _Harvey P. Dale_, Aug 10 2013 *)
%o (Haskell)
%o a144328 n k = a144328_tabl !! (n-1) !! (k-1)
%o a144328_row n = a144328_tabl !! (n-1)
%o a144328_tabl = [1] : map (\xs@(x:_) -> x : xs) a002260_tabl
%o -- _Reinhard Zumkeller_, Apr 29 2015
%Y Cf. A000124, A000142, A002260, A064999, A028310, A156702.
%K nonn,tabl
%O 1,6
%A _Gary W. Adamson_, Sep 18 2008
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