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A117178 Riordan array (c(x^2)/sqrt(1-4*x^2), x*c(x^2)), c(x) the g.f. of A000108.

%I A117178 #14 Aug 08 2022 23:56:01
%S A117178 1,0,1,3,0,1,0,4,0,1,10,0,5,0,1,0,15,0,6,0,1,35,0,21,0,7,0,1,0,56,0,
%T A117178 28,0,8,0,1,126,0,84,0,36,0,9,0,1,0,210,0,120,0,45,0,10,0,1,462,0,330,
%U A117178 0,165,0,55,0,11,0,1,0,792,0,495,0,220,0,66,0,12,0,1
%N A117178 Riordan array (c(x^2)/sqrt(1-4*x^2), x*c(x^2)), c(x) the g.f. of A000108.
%C A117178 Row sums are A058622(n+1). Diagonal sums are A001791(n+1), with interpolated zeros. Inverse is A117179.
%C A117178 De-aerated and rows reversed, this matrix apparently becomes A014462. The nonzero antidiagonals are embedded in several entries and apparently contain partial sums of previous nonzero antidiagonals. - _Tom Copeland_, May 30 2017
%H A117178 G. C. Greubel, <a href="/A117178/b117178.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A117178 T(n,k) = C(n+1, (n-k)/2) * (1 + (-1)^(n-k))/2.
%F A117178 Column k has e.g.f. Bessel_I(k,2x) + Bessel_I(k+2, 2x).
%F A117178 From _G. C. Greubel_, Aug 08 2022: (Start)
%F A117178 Sum_{k=0..n} T(n, k) = A058622(n+1).
%F A117178 Sum_{k=0..floor(n/2)} T(n-k, k) = ((1+(-1)^n)/2) * A001791((n+2)/2).
%F A117178 T(2*n, n) = ((1+(-1)^n)/2) * A052203(n/2).
%F A117178 T(2*n+1, n) = ((1-(-1)^n)/2) * A224274((n+1)/2).
%F A117178 T(2*n-1, n-1) = ((1+(-1)^n)/2) * A224274(n/2). (End)
%e A117178 Triangle begins
%e A117178     1;
%e A117178     0,  1;
%e A117178     3,  0,  1;
%e A117178     0,  4,  0,  1;
%e A117178    10,  0,  5,  0,  1;
%e A117178     0, 15,  0,  6,  0,  1;
%e A117178    35,  0, 21,  0,  7,  0,  1;
%e A117178     0, 56,  0, 28,  0,  8,  0,  1;
%e A117178   126,  0, 84,  0, 36,  0,  9,  0,  1;
%t A117178 T[n_, k_]:= Binomial[n+1, (n-k)/2]*(1+(-1)^(n-k))/2;
%t A117178 Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Aug 08 2022 *)
%o A117178 (Magma) [(1+(-1)^(n-k))*Binomial(n+1, Floor((n-k)/2))/2: k in [0..n], n in [0..15]]; // _G. C. Greubel_, Aug 08 2022
%o A117178 (SageMath)
%o A117178 def A117178(n,k): return (1 + (-1)^(n-k))*binomial(n+1, (n-k)//2)/2
%o A117178 flatten([[A117178(n,k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Aug 08 2022
%Y A117178 Cf. A001791, A014462, A052203, A058622 (row sums), A117179, A224274.
%K A117178 easy,nonn,tabl
%O A117178 0,4
%A A117178 _Paul Barry_, Mar 01 2006