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A065602 Triangle T(n,k) giving number of hill-free Dyck paths of length 2n and having height of first peak equal to k.

%I A065602 #28 Jun 01 2022 09:44:03
%S A065602 1,1,1,3,2,1,8,6,3,1,24,18,10,4,1,75,57,33,15,5,1,243,186,111,54,21,6,
%T A065602 1,808,622,379,193,82,28,7,1,2742,2120,1312,690,311,118,36,8,1,9458,
%U A065602 7338,4596,2476,1164,474,163,45,9,1,33062,25724,16266,8928,4332,1856,692,218,55,10,1
%N A065602 Triangle T(n,k) giving number of hill-free Dyck paths of length 2n and having height of first peak equal to k.
%C A065602 A Riordan triangle.
%C A065602 Subtriangle of triangle in A167772. - _Philippe Deléham_, Nov 14 2009
%C A065602 Riordan array (f(x), x*g(x)) where f(x) is the g.f. of A000958 and g(x) is the g.f. of A000108. - _Philippe Deléham_, Jan 23 2010
%H A065602 Reinhard Zumkeller, <a href="/A065602/b065602.txt">Rows n = 2..125 of triangle, flattened</a>
%H A065602 Emeric Deutsch and L. Shapiro, <a href="https://doi.org/10.1016/S0012-365X(01)00121-2">A survey of the Fine numbers</a>, Discrete Math., 241 (2001), 241-265.
%F A065602 T(n, 2) = A000958(n-1).
%F A065602 Sum_{k=2..n} T(n, k) = A000957(n+1).
%F A065602 From _Emeric Deutsch_, Feb 23 2004: (Start)
%F A065602 T(n, k) = Sum_{j=0..floor((n-k)/2)} (k-1+2*j)*binomial(2*n-k-1-2*j, n-1)/(2*n-k-1-2*j).
%F A065602 G.f.: t^2*z^2*C/( (1-z^2*C^2)*(1-t*z*C) ), where C = (1-sqrt(1-4*z))/(2*z) is the Catalan function. (End)
%F A065602 T(n,k) = A167772(n-1,k-1), k=2..n. - _Reinhard Zumkeller_, May 15 2014
%F A065602 From _G. C. Greubel_, May 26 2022: (Start)
%F A065602 T(n, n-1) = A000027(n-2).
%F A065602 T(n, n-2) = A000217(n-2).
%F A065602 T(n, n-3) = A166830(n-3). (End)
%e A065602 T(3,2)=1 reflecting the unique Dyck path (UUDUDD) of length 6, with no hills and height of first peak equal to 2.
%e A065602 Triangle begins:
%e A065602      1;
%e A065602      1,    1;
%e A065602      3,    2,    1;
%e A065602      8,    6,    3,   1;
%e A065602     24,   18,   10,   4,   1;
%e A065602     75,   57,   33,  15,   5,   1;
%e A065602    243,  186,  111,  54,  21,   6,  1;
%e A065602    808,  622,  379, 193,  82,  28,  7,  1;
%e A065602   2742, 2120, 1312, 690, 311, 118, 36,  8,  1;
%p A065602 a := proc(n,k) if n=0 and k=0 then 1 elif k<2 or k>n then 0 else sum((k-1+2*j)*binomial(2*n-k-1-2*j,n-1)/(2*n-k-1-2*j),j=0..floor((n-k)/2)) fi end: seq(seq(a(n,k),k=2..n),n=1..14);
%t A065602 nmax = 12; t[n_, k_] := Sum[(k-1+2j)*Binomial[2n-k-1-2j, n-1] / (2n-k-1-2j), {j, 0, (n-k)/2}]; Flatten[ Table[t[n, k], {n, 2, nmax}, {k, 2, n}]] (* _Jean-François Alcover_, Nov 08 2011, after Maple *)
%o A065602 (Haskell)
%o A065602 a065602 n k = sum
%o A065602    [(k-1+2*j) * a007318' (2*n-k-1-2*j) (n-1) `div` (2*n-k-1-2*j) |
%o A065602     j <- [0 .. div (n-k) 2]]
%o A065602 a065602_row n = map (a065602 n) [2..n]
%o A065602 a065602_tabl = map a065602_row [2..]
%o A065602 -- _Reinhard Zumkeller_, May 15 2014
%o A065602 (SageMath)
%o A065602 def T(n,k): return sum( (k+2*j-1)*binomial(2*n-2*j-k-1, n-1)/(2*n-2*j-k-1) for j in (0..(n-k)//2) )
%o A065602 flatten([[T(n,k) for k in (2..n)] for n in (2..12)]) # _G. C. Greubel_, May 26 2022
%Y A065602 Row sums give A000957 (the Fine sequence).
%Y A065602 First column is A000958.
%Y A065602 Cf. A000108, A007318, A167772.
%K A065602 nonn,tabl,easy,nice
%O A065602 2,4
%A A065602 _N. J. A. Sloane_, Dec 02 2001
%E A065602 More terms from _Emeric Deutsch_, Feb 23 2004