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%I #27 Jul 27 2019 21:50:21
%S 1,1,2,1,4,5,1,6,15,14,1,8,30,56,42,1,10,50,140,210,132,1,12,75,280,
%T 630,792,429,1,14,105,490,1470,2772,3003,1430,1,16,140,784,2940,7392,
%U 12012,11440,4862,1,18,180,1176,5292,16632,36036,51480,43758,16796,1,20,225
%N Triangle read by rows: T(n,k) = binomial(2k+2,k+1)*binomial(n,k)/(k+2) (0 <= k <= n).
%C Also, with offset 1, triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and ending at the point (2k,0) (1 <= k <= n). A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of a path is defined to be the number of its steps. For example, T(3,2)=4 because we have UDUUDL, UUUDLD, UUDUDL and UUUDDL.
%C This sequence gives the coefficients of the Jensen polynomials (increasing powers of x) of degree n and shift 1 for the Catalan sequence A000108. See A098474 for a similar comment. - _Wolfdieter Lang_, Jun 25 2019
%H E. Deutsch, E. Munarini, S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.jspi.2010.01.015">Skew Dyck paths</a>, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
%F Sum of row n = A002212(n+1).
%F T(n,n) = Catalan(n+1) (A000108).
%F Sum_{k=1..n} k*T(n,k) = A026388(n).
%F With offset 1, T(n,k) = c(k)*binomial(n-1,k-1), where c(j) = binomial(2j,j)/(j+1) is a Catalan number (A000108).
%F G.f.: G-1, where G=G(t,z) satisfies G = 1 + t*z*G^2 + z*(G-1).
%F T(n, k) = 4^k*[x^k]hypergeometric([3/2, -n], [3], -x). - _Peter Luschny_, Feb 03 2015, based on an observation of _Peter Bala_ in A254632.
%F T(n, k) = (-1)^k*Catalan(k+1)*Pochhammer(-n,k)/k!. - _Peter Luschny_, Feb 05 2015
%e Triangle begins:
%e 1;
%e 1, 2;
%e 1, 4, 5;
%e 1, 6, 15, 14;
%e 1, 8, 30, 56, 42;
%e 1, 10, 50, 140, 210, 132;
%e 1, 12, 75, 280, 630, 792, 429;
%p T:=(n,k)->binomial(2*k+2,k+1)*binomial(n,k)/(k+2): for n from 0 to 10 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
%p h := n -> simplify(hypergeom([3/2, -n], [3], -x)):
%p seq(print(seq(4^k*coeff(h(n), x, k), k=0..n)), n=0..9); # _Peter Luschny_, Feb 03 2015
%t Flatten[Table[Binomial[2k+2,k+1] Binomial[n,k]/(k+2),{n,0,10},{k,0,n}]] (* _Harvey P. Dale_, Jul 20 2013 *)
%o (Sage)
%o def A108198(n,k):
%o return (-1)^k*catalan_number(k+1)*rising_factorial(-n,k)/factorial(k)
%o for n in range(7): [A108198(n,k) for k in (0..n)] # _Peter Luschny_, Feb 05 2015
%Y Mirror image of A126181.
%Y Cf. A000108, A002212, A026388, A098474.
%K nonn,tabl,easy
%O 0,3
%A _Emeric Deutsch_, Jun 15 2005, Mar 30 2007
%E Edited by _N. J. A. Sloane_ at the suggestion of _Andrew Plewe_, Jun 16 2007
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