This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A033877 #119 Jul 31 2024 09:08:15
%S A033877 1,1,2,1,4,6,1,6,16,22,1,8,30,68,90,1,10,48,146,304,394,1,12,70,264,
%T A033877 714,1412,1806,1,14,96,430,1408,3534,6752,8558,1,16,126,652,2490,7432,
%U A033877 17718,33028,41586,1,18,160,938,4080,14002,39152,89898,164512,206098
%N A033877 Triangular array read by rows associated with Schroeder numbers: T(1,k) = 1; T(n,k) = 0 if k < n; T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k).
%C A033877 A106579 is in some ways a better version of this sequence, but since this was entered first it will be the main entry for this triangle.
%C A033877 The diagonals of this triangle are self-convolutions of the main diagonal A006318: 1, 2, 6, 22, 90, 394, 1806, ... . - _Philippe Deléham_, May 15 2005
%C A033877 From _Johannes W. Meijer_, Sep 22 2010, Jul 15 2013: (Start)
%C A033877 Note that for the terms T(n,k) of this triangle n indicates the column and k the row.
%C A033877 The triangle sums, see A180662, link Schroeder's triangle with several sequences, see the crossrefs. The mirror of this triangle is A080247.
%C A033877 Quite surprisingly the Kn1p sums, p >= 1, are all related to A026003 and crystal ball sequences for n-dimensional cubic lattices (triangle offset is 0): Kn11(n) = A026003(n), Kn12(n) = A026003(n+2) - 1, Kn13(n) = A026003(n+4) - A005408(n+3), Kn14(n) = A026003(n+6) - A001844(n+4), Kn15(n) = A026003(n+8) - A001845(n+5), Kn16(n) = A026003(n+10) - A001846(n+6), Kn17(n) = A026003(n+12) - A001847(n+7), Kn18(n) = A026003(n+14) - A001848(n+8), Kn19(n) = A026003(n+16) - A001849(n+9), Kn110(n) = A026003(n+18) - A008417(n+10), Kn111(n) = A026003(n+20) - A008419(n+11), Kn112(n) = A026003(n+22) - A008421(n+12). (End)
%C A033877 T(n,k) is the number of normal semistandard Young tableaux with two columns, one of height k and one of height n. The recursion can be seen by performing jeu de taquin deletion on all instances of the smallest value. (If there are two instances of the smallest value, jeu de taquin deletion will always shorten the right column first and the left column second.) - _Jacob Post_, Jun 19 2018
%H A033877 T. D. Noe, <a href="/A033877/b033877.txt">Rows k = 1..50 of triangle, flattened</a>
%H A033877 Henry Bottomley, <a href="/A001003/a001003.gif">Illustration of initial terms</a>
%H A033877 Kevin Brown, <a href="http://www.mathpages.com/home/kmath397/kmath397.htm">Hipparchus on Compound Statements</a>, 1994-2010. - _Johannes W. Meijer_, Sep 22 2010
%H A033877 James East and Nicholas Ham, <a href="https://arxiv.org/abs/1811.05735">Lattice paths and submonoids of Z^2</a>, arXiv:1811.05735 [math.CO], 2018.
%H A033877 G. Kreweras, <a href="http://www.numdam.org/item?id=BURO_1973__20__3_0">Sur les hiérarchies de segments</a>, Cahiers Bureau Universitaire Recherche Opérationnelle, Cahier 20, Inst. Statistiques, Univ. Paris, 1973.
%H A033877 G. Kreweras, <a href="/A001844/a001844.pdf">Sur les hiérarchies de segments</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
%H A033877 G. Kreweras, <a href="/A006318/a006318_2.pdf">Aires des chemins surdiagonaux et application à un problème économique</a>, Cahiers du Bureau universitaire de recherche opérationnelle Série Recherche 24 (1976): 1-8. [Annotated scanned copy]
%H A033877 J. W. Meijer, <a href="http://www.scielo.org.bo/scielo.php?script=sci_arttext&pid=S1683-07892010000200009">Famous numbers on a chessboard</a>, Acta Nova, Volume 4, No.4, December 2010. pp. 589-598.
%H A033877 J. M. Oh, <a href="http://ijfs.usb.ac.ir/article_1335_40.html">An explicit formula for the number of fuzzy subgroups of a finite abelian p-group of rank two</a>, Iranian Journal of Fuzzy Systems, Dec 2013, Vol. 10 Issue 6, pp. 125-135.
%H A033877 E. Pergola and R. A. Sulanke, <a href="https://cs.uwaterloo.ca/journals/JIS/PergolaSulanke/">Schroeder Triangles, Paths and Parallelogram Polyominoes</a>, J. Integer Sequences, 1 (1998), #98.1.7.
%H A033877 S. Samieinia, <a href="http://dx.doi.org/10.4171/PM/1858">The number of continuous curves in digital geometry</a>, Port. Math. 67 (1) (2010) 75-89, last table.
%H A033877 R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Sulanke/delannoy.html">Objects counted by the central Delannoy numbers</a>, J. Integer Seq. 6 (2003), Article 03.1.5, 19 pp.
%H A033877 Luis Verde-Star, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Verde/verde4.html">A Matrix Approach to Generalized Delannoy and Schröder Arrays</a>, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.
%F A033877 As an upper right triangle: a(n, k) = a(n, k-1) + a(n-1, k-1) + a(n-1, k) if k >= n >= 0 and a(n, k) = 0 otherwise.
%F A033877 G.f.: Sum T(n, k)*x^n*y^k = (1-x*y-(x^2*y^2-6*x*y+1)^(1/2)) / (x*(2*y+x*y-1+(x^2*y^2-6*x*y+1)^(1/2))). - _Vladeta Jovovic_, Feb 16 2003
%F A033877 Another version of A000007 DELTA [0, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...] = 1, 1, 0, 1, 2, 0, 1, 4, 6, 0, 1, 6, 16, 22, 0, 1, ..., where DELTA is Deléham's operator defined in A084938.
%F A033877 Sum_{n=1..floor((k+1)/2)} T(n+p-1, k-n+p) = A026003(2*p+k-3) - A008288(2*p+k-3, p-2), p >= 2, k >= 1. - _Johannes W. Meijer_, Sep 28 2013
%F A033877 From _G. C. Greubel_, Mar 23 2023: (Start)
%F A033877 (t(n, k) as a lower triangle)
%F A033877 t(n, k) = t(n, k-1) + t(n-1, k-1) + t(n-1, k) with t(n, 1) = 1.
%F A033877 t(n, n) = A006318(n-1).
%F A033877 t(2*n-1, n) = A330801(n-1).
%F A033877 t(2*n-2, n) = A103885(n-1), n > 1.
%F A033877 Sum_{k=1..n-1} t(n, k) = A238112(n), n > 1.
%F A033877 Sum_{k=1..n} t(n, k) = A001003(n).
%F A033877 Sum_{k=1..n-1} (-1)^(k-1)*t(n, k) = (-1)^n*A001003(n-1), n > 1.
%F A033877 Sum_{k=1..n} (-1)^(k-1)*t(n, k) = A080243(n-1).
%F A033877 Sum_{k=1..floor((n+1)/2)} t(n-k+1, k) = A026003(n-1). (End)
%e A033877 Triangle starts:
%e A033877 1;
%e A033877 1, 2;
%e A033877 1, 4, 6;
%e A033877 1, 6, 16, 22;
%e A033877 1, 8, 30, 68, 90;
%e A033877 1, 10, 48, 146, 304, 394;
%e A033877 1, 12, 70, 264, 714, 1412, 1806;
%e A033877 ... - _Joerg Arndt_, Sep 29 2013
%p A033877 T := proc(n, k) option remember; if n=1 then return(1) fi; if k<n then return(0) fi; T(n, k-1)+T(n-1, k-1)+T(n-1, k) end: seq(seq(T(n,k), n = 1..k), k=1..10); # _Johannes W. Meijer_, Sep 22 2010, revised Jul 17 2013
%t A033877 T[1, _]:= 1; T[n_, k_]/;(k<n):= 0; T[n_, k_]:= T[n, k]= T[n, k-1] +T[n-1, k-1] + T[n-1, k]; Table[T[k,n], {n,15}, {k,n}]//Flatten
%o A033877 (Haskell)
%o A033877 a033877 n k = a033877_tabl !! n !! k
%o A033877 a033877_row n = a033877_tabl !! n
%o A033877 a033877_tabl = iterate
%o A033877 (\row -> scanl1 (+) $ zipWith (+) ([0] ++ row) (row ++ [0])) [1]
%o A033877 -- _Reinhard Zumkeller_, Apr 17 2013
%o A033877 (Sage)
%o A033877 def A033877_row(n):
%o A033877 @cached_function
%o A033877 def prec(n, k):
%o A033877 if k==n: return 1
%o A033877 if k==0: return 0
%o A033877 return prec(n-1,k-1)-2*sum(prec(n,k+i-1) for i in (2..n-k+1))
%o A033877 return [(-1)^k*prec(n, n-k) for k in (0..n-1)]
%o A033877 for n in (1..10): print(A033877_row(n)) # _Peter Luschny_, Mar 16 2016
%o A033877 (SageMath)
%o A033877 @CachedFunction
%o A033877 def t(n, k): # t = A033847
%o A033877 if (k<0 or k>n): return 0
%o A033877 elif (k==1): return 1
%o A033877 else: return t(n, k-1) + t(n-1, k-1) + t(n-1, k)
%o A033877 flatten([[t(n,k) for k in range(1,n+1)] for n in range(1, 16)]) # _G. C. Greubel_, Mar 23 2023
%o A033877 (Magma)
%o A033877 function t(n,k)
%o A033877 if k le 0 or k gt n then return 0;
%o A033877 elif k eq 1 then return 1;
%o A033877 else return t(n,k-1) + t(n-1,k-1) + t(n-1,k);
%o A033877 end if;
%o A033877 end function;
%o A033877 [t(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Mar 23 2023
%Y A033877 Essentially same triangle as A080247 and A080245 but with rows read in reversed order. Also essentially the same triangle as A106579.
%Y A033877 Cf. A000007, A006318, A006319, A006320, A006321, A008288, A080243.
%Y A033877 Cf. A084938, A103885, A238112, A330801.
%Y A033877 Cf. A001003 (row sums), A026003 (antidiagonal sums).
%Y A033877 Triangle sums (see the comments): A001003 (Row1, Row2), A026003 (Kn1p, p >= 1), A006603 (Kn21), A227504 (Kn22), A227505 (Kn23), A006603(2*n) (Kn3), A001850 (Kn4), A227506 (Fi1), A010683 (Fi2).
%K A033877 nonn,tabl,nice
%O A033877 1,3
%A A033877 _N. J. A. Sloane_
%E A033877 More terms from _David W. Wilson_