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A126325 Triangle read by rows: T(n,k) = binomial(2*n+1, n-k) (1 <= k <= n).

%I A126325 #27 Jul 26 2023 21:11:12
%S A126325 1,5,1,21,7,1,84,36,9,1,330,165,55,11,1,1287,715,286,78,13,1,5005,
%T A126325 3003,1365,455,105,15,1,19448,12376,6188,2380,680,136,17,1,75582,
%U A126325 50388,27132,11628,3876,969,171,19,1,293930,203490,116280,54264,20349,5985,1330,210,21,1
%N A126325 Triangle read by rows: T(n,k) = binomial(2*n+1, n-k) (1 <= k <= n).
%C A126325 T(n,k) is the total area between the lines y=k-1 and y=k in all Dyck paths of semilength n (1 <= k <= n).
%C A126325 With row and column indices starting at 0, this triangle is the Riordan array ( c(x)^4/(2 - c(x)), x*c^2(x) ), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. - _Peter Bala_, Mar 12 2022
%C A126325 Equals A111418 when k starts at 0. - _Georg Fischer_, Jul 26 2023
%H A126325 G. C. Greubel, <a href="/A126325/b126325.txt">Rows n = 1..100 of triangle, flattened</a>
%F A126325 T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + T(n-1,k+1) for n >= 2, k >= 2.
%F A126325 T(n,1) = A002054(n); T(n,2) = A003516(n); T(n,3) = A030053(n);
%F A126325 T(n,4) = A030054(n); T(n,5) = A030055(n).
%F A126325 Row sums yield A008549.
%e A126325 Triangle begins:
%e A126325      1;
%e A126325      5,    1;
%e A126325     21,    7,    1;
%e A126325     84,   36,    9,    1;
%e A126325    330,  165,   55,   11,    1;
%e A126325   1287,  715,  286,   78,   13,    1;
%e A126325   5005, 3003, 1365,  455,  105,   15,    1;
%e A126325   ..
%p A126325 T:=(n,k)->binomial(2*n+1,n-k): for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
%t A126325 t[n_, k_] := Binomial[2n + 1, n - k]; Table[ t[n, k], {n, 10}, {k, n}] // Flatten
%o A126325 (PARI) for(n=1,15, for(k=1,n, print1(binomial(2*n+1, n-k), ", "))) \\ _G. C. Greubel_, Oct 23 2018
%o A126325 (Magma) [[Binomial(2*n+1, n-k): k in [1..n]]: n in [1..15]]; // _G. C. Greubel_, Oct 23 2018
%o A126325 (GAP) T:=Flat(List([1..10],n->List([1..n],k->Binomial(2*n+1,n-k)))); # _Muniru A Asiru_, Oct 24 2018
%Y A126325 Cf. A000108, A008549, A002054, A003516, A030053, A030054, A030055, A111418.
%K A126325 nonn,tabl
%O A126325 1,2
%A A126325 _Emeric Deutsch_, Mar 11 2007