cp's OEIS Frontend

A132812 Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).

%I A132812 #32 Oct 22 2021 23:35:39
%S A132812 1,2,2,3,9,3,4,24,24,4,5,50,100,50,5,6,90,300,300,90,6,7,147,735,1225,
%T A132812 735,147,7,8,224,1568,3920,3920,1568,224,8,9,324,3024,10584,15876,
%U A132812 10584,3024,324,9,10,450,5400,25200,52920,52920,25200,5400,450,10
%N A132812 Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
%C A132812 A127648 * A001263. (Original name by Gary W. Adamson.)
%C A132812 Let a meander be defined as in the link and m = 2. Then T(n,k) counts the invertible meanders of length m(n+1) built from arcs with central angle 360/m whose binary representation have mk '1's. - _Peter Luschny_, Dec 19 2011
%C A132812 Antidiagonal sums = A110320. - _Philippe Deléham_, Jun 08 2013
%H A132812 Michael De Vlieger, <a href="/A132812/b132812.txt">Table of n, a(n) for n = 1..11325</a> (rows 1..150).
%H A132812 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/Meander">Meanders</a>.
%F A132812 A127648 * A001263 as infinite lower triangular matrices.
%F A132812 a(n) = n * A001263(n,k).
%F A132812 T(n,k) = binomial(n,k)*binomial(n,k-1). - _Philippe Deléham_, Jun 08 2013
%F A132812 G.f.: x*d(N(x,y))/dx, where N(x,y) is g.f. for Narayana numbers A001263. - _Vladimir Kruchinin_, Oct 22 2021
%e A132812 First few rows of the triangle are:
%e A132812   1;
%e A132812   2, 2;
%e A132812   3, 9, 3;
%e A132812   4, 24, 24, 4;
%e A132812   5, 50, 100, 50, 5;
%e A132812   6, 90, 300, 300, 90, 6;
%e A132812   ...
%e A132812 Row 4 = (4, 24, 24, 4) = 4 * (1, 6, 6, 1), where (1, 6, 6, 1) = row 4 of the Narayana triangle. - _Gary W. Adamson_
%e A132812 T(3,1) = 3 because the invertible meanders of length 8 and central angle 180 degree which have two '1's in their binary representation are {10000100, 10010000, 11000000}. - _Peter Luschny_, Dec 19 2011
%p A132812 A132812 := (n,k) -> k*binomial(n,k)^2/(n-k+1);
%p A132812 seq(print(seq(A132812(n,k),k=0..n-1)),n=1..6); # _Peter Luschny_, Dec 19 2011
%t A132812 Table[k Binomial[n, k]^2/(n - k + 1), {n, 10}, {k, n}] // Flatten (* _Michael De Vlieger_, Nov 15 2017 *)
%Y A132812 Row sums: A001791. Cf. A001263, A127648, A001791, A202409.
%K A132812 nonn,tabl
%O A132812 1,2
%A A132812 _Gary W. Adamson_, Sep 01 2007
%E A132812 New name from _Peter Luschny_, Dec 19 2011
%E A132812 a(53) corrected by _Michael De Vlieger_, Nov 15 2017