# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: id:a093127
Showing 1-1 of 1

%I A093127 #25 Sep 08 2022 08:45:13
%S A093127 1,1,1,2,1,5,1,9,3,1,14,14,1,20,40,4,1,27,90,30,1,35,175,125,5,1,44,
%T A093127 308,385,55,1,54,504,980,315,6,1,65,780,2184,1274,91,1,77,1155,4410,
%U A093127 4116,686,7,1,90,1650,8250,11340,3528,140,1,104,2288,14520,27720,14112,1344,8
%N A093127 Triangle read by rows: differences of Narayana numbers.
%C A093127 T(n,k) (0 <= k <= n/2) is the number of dissections of a regular (n+2)-gon using k strictly disjoint diagonals (diagonals join nonconsecutive vertices and strictly disjoint means no two cross or share an endpoint).
%C A093127 Row n contains 1 + floor(n/2) entries. - _Emeric Deutsch_, Sep 18 2014
%C A093127 T(n,k) is the number of paths in B(n+1) that do not start with H and have k steps of weight 2 (i.e., H or u). Example: T(3,1) = 5 because the paths in B(4) that do not start with H are hhhh, hHh, hhH, uhd, hud, and udh; the last five contain exactly 1 step of weight 2. - _Emeric Deutsch_, Sep 18 2014
%C A093127 B(n) is the set of lattice paths of weight n that start in (0,0), end on the horizontal axis and never go below this axis, whose steps are of the following four kinds: h = (1,0) of weight 1, H = (1,0) of weight 2, u = (1,1) of weight 2, and d = (1,-1) of weight 1. The weight of a path is the sum of the weights of its steps. - _Emeric Deutsch_, Sep 18 2014
%H A093127 G. C. Greubel, <a href="/A093127/b093127.txt">Rows n = 0..100 of triangle, flattened</a>
%H A093127 M. Bona and A. Knopfmacher, <a href="http://dx.doi.org/10.1007/s00026-010-0060-7">On the probability that certain compositions have the same number of parts</a>, Ann. Comb., 14 (2010), 291-306. - _Emeric Deutsch_, Sep 18 2014
%F A093127 T(n, k) = Narayana(n-k+2, k+1) - Narayana(n-k+1, k) where Narayana(n, k) = binomial(n, k)*binomial(n, k-1)/n is A001263.
%F A093127 G.f.: G=G(t,z) satisfies t*z^4*G^2 - (1 - z - 2*t*z^2 - t*z^3 + t^2*z^4)*G + 1 = 0. - _Emeric Deutsch_, Sep 18 2014
%e A093127 T(3,1) = 5 because there are 5 ways to insert a single diagonal into a pentagon.
%e A093127 Triangle begins:
%e A093127   1;
%e A093127   1;
%e A093127   1,  2;
%e A093127   1,  5;
%e A093127   1,  9,  3;
%e A093127   1, 14, 14;
%e A093127   1, 20, 40,  4;
%e A093127   1, 27, 90, 30;
%p A093127 eq := t*z^4*G^2-(1-z-2*t*z^2-t*z^3+t^2*z^4)*G+1 = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 18)): for n from 0 to 15 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 15 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form # _Emeric Deutsch_, Sep 18 2014
%p A093127 T := proc (n, k) if (1/2)*n+1/2 < k then 0 else binomial(n-k+2, k+1)*binomial(n-k+2, k)/(n-k+2)-binomial(n-k+1, k)*binomial(n-k+1, k-1)/(n-k+1) end if end proc: for n from 0 to 15 do seq(T(n, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form # _Emeric Deutsch_, Sep 18 2014
%t A093127 Nara[n_, k_]:= Binomial[n, k]*Binomial[n, k-1]/n; T[n_, k_]:= Nara[n-k+2, k+1] - Nara[n-k+1, k]; Table[T[n, k], {n,0,15}, {k,0,Floor[n/2]}]//Flatten (* _G. C. Greubel_, Dec 28 2019 *)
%o A093127 (PARI) T(n,k) = my(b=binomial); b(n-k+2,k+1)*b(n-k+2,k)/(n-k+2) - b(n-k+1,k)* b(n-k+1,k-1)/(n-k+1);
%o A093127 for(n=0,15, for(k=0, n\2, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Dec 28 2019
%o A093127 (Magma) B:=Binomial; T:= func< n, k | B(n-k+2,k+1)*B(n-k+2,k)/(n-k+2) - B(n-k+1,k)*B(n-k+1,k-1)/(n-k+1) >;
%o A093127 [T(n,k): k in [0..Floor(n/2)], n in [0..15]]; // _G. C. Greubel_, Dec 28 2019
%o A093127 (Sage)
%o A093127 b=binomial;
%o A093127 def T(n,k): return b(n-k+2,k+1)*b(n-k+2,k)/(n-k+2) - b(n-k+1,k)* b(n-k+1,k-1)/(n-k+1)
%o A093127 [[T(n,k) for k in (0..floor(n/2))] for n in (0..15)] # _G. C. Greubel_, Dec 28 2019
%o A093127 (GAP)
%o A093127 B:=Binomial;;
%o A093127 T:= function(n,k)
%o A093127     return B(n-k+2,k+1)*B(n-k+2,k)/(n-k+2) - B(n-k+1,k)*B(n-k+1,k-1)/(n-k+1);
%o A093127     end;
%o A093127 Flat(List([0..15], n-> List([0..Int(n/2)], k-> T(n,k) ))); # _G. C. Greubel_, Dec 28 2019
%K A093127 easy,nonn,tabf
%O A093127 0,4
%A A093127 _David Callan_, Mar 23 2004

# Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE