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A093768 Positive first differences of the rows of triangle A088459, which enumerates symmetric Dyck paths.

%I A093768 #15 Feb 03 2025 12:06:19
%S A093768 1,1,1,1,2,3,1,3,8,6,1,4,15,20,20,1,5,24,45,75,50,1,6,35,84,189,210,
%T A093768 175,1,7,48,140,392,588,784,490,1,8,63,216,720,1344,2352,2352,1764,1,
%U A093768 9,80,315,1215,2700,5760,7560,8820,5292,1,10,99,440,1925,4950,12375,19800
%N A093768 Positive first differences of the rows of triangle A088459, which enumerates symmetric Dyck paths.
%C A093768 Suggested by Bozydar Dubalski (slawb(AT)atr.bydgoszcz.pl). Related to walks on a square lattice: main diagonal forms A005558, secondary diagonals form A005559, A005560, A005561, A005562, A005563.
%C A093768 Apparently row-reversed version of A052174. - _R. J. Mathar_, Feb 03 2025
%H A093768 G. C. Greubel, <a href="/A093768/b093768.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A093768 T(n, k) = C(n+1, ceiling(k/2))*C(n, floor(k/2)) - C(n+1, ceiling((k-1)/2))*C(n, floor((k-1)/2)) for n>=k>=0.
%e A093768 1;
%e A093768 1, 1;
%e A093768 1, 2, 3;
%e A093768 1, 3, 8, 6;
%e A093768 1, 4, 15, 20, 20;
%e A093768 1, 5, 24, 45, 75, 50;
%e A093768 1, 6, 35, 84, 189, 210, 175;
%p A093768 A093768 := proc(n,k)
%p A093768     if k = 0 then
%p A093768         A088459(n,k);
%p A093768     else
%p A093768         A088459(n,k)-A088459(n,k-1);
%p A093768     end if;
%p A093768 end proc:
%p A093768 seq(seq(A093768(n,k),k=0..n-1),n=1..10) ; # _R. J. Mathar_, Apr 02 2017
%t A093768 T[n_, k_] := Binomial[n + 1, Ceiling[k/2]]*Binomial[n, Floor[k/2]] - Binomial[n + 1, Ceiling[(k - 1)/2]]*Binomial[n, Floor[(k - 1)/2]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _G. C. Greubel_, Oct 25 2017 *)
%o A093768 (PARI) {T(n,k) =binomial(n+1,ceil(k/2))*binomial(n,floor(k/2)) -binomial(n+1,ceil((k-1)/2))*binomial(n,floor((k-1)/2))}
%Y A093768 Cf. A088459, A005558-A005562, A005563 (column 3), A005564 (column 4), A005565 (column 5), A005566 (row sums).
%K A093768 nonn,tabl
%O A093768 0,5
%A A093768 _Paul D. Hanna_, Apr 16 2004