|
%I #41 Jan 23 2023 12:06:50
%S 1,1,1,1,1,2,1,1,3,3,1,1,4,4,6,1,1,5,5,10,10,1,1,6,6,15,15,20,1,1,7,7,
%T 21,21,35,35,1,1,8,8,28,28,56,56,70,1,1,9,9,36,36,84,84,126,126,1,1,
%U 10,10,45,45,120,120,210,210,252,1,1,11,11,55,55,165,165,330,330,462,462,1
%N Triangle read by rows: row n is row n of Pascal's triangle (A007318) sorted into increasing order.
%C By rows, equals partial sums of A053121 reversed rows. Example: Row 4 of A053121 = (2, 0, 3, 0, 1) -> (1, 0, 3, 0, 2) -> (1, 1, 4, 4, 6). - _Gary W. Adamson_, Dec 28 2008, edited by _Michel Marcus_, Sep 22 2015
%H Reinhard Zumkeller, <a href="/A107430/b107430.txt">Rows n = 0..125 of triangle, flattened</a>
%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F T(n,k) = C(n,floor(k/2)). - _Paul Barry_, Dec 15 2006; corrected by _Philippe Deléham_, Mar 15 2007
%F Sum_{k=0..n} T(n,k)*x^(n-k) = A127363(n), A127362(n), A127361(n), A126869(n), A001405(n), A000079(n), A127358(n), A127359(n), A127360(n) for x=-4,-3,-2,-1,0,1,2,3,4 respectively. - _Philippe Deléham_, Mar 29 2007
%e Triangle begins:
%e 1;
%e 1,1;
%e 1,1,2;
%e 1,1,3,3;
%e 1,1,4,4,6;
%p for n from 0 to 10 do sort([seq(binomial(n,k),k=0..n)]) od; # yields sequence in triangular form. - _Emeric Deutsch_, May 28 2005
%t Flatten[ Table[ Sort[ Table[ Binomial[n, k], {k, 0, n}]], {n, 0, 12}]] (* _Robert G. Wilson v_, May 28 2005 *)
%o (Haskell)
%o import Data.List (sort)
%o a107430 n k = a107430_tabl !! n !! k
%o a107430_row n = a107430_tabl !! n
%o a107430_tabl = map sort a007318_tabl
%o -- _Reinhard Zumkeller_, May 26 2013
%o (Magma) /* As triangle */ [[Binomial(n,Floor(k/2)) : k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Sep 22 2015
%o (PARI) for(n=0,20, for(k=0,n, print1(binomial(n,floor(k/2)), ", "))) \\ _G. C. Greubel_, May 22 2017
%Y A061554 is similar but with rows sorted into decreasing order.
%Y Cf. A034868.
%Y Cf. A053121. - _Gary W. Adamson_, Dec 28 2008
%Y Cf. A103284.
%K nonn,tabl,easy
%O 0,6
%A _Philippe Deléham_, May 21 2005
%E More terms from _Emeric Deutsch_ and _Robert G. Wilson v_, May 28 2005
|
